Test papers. Grade 10
Test work on the topic “Kinematics of a material point.”
A basic level of
Option 1
A1. The trajectory of a moving material point in a finite time is
line segment
part of the plane
finite set of points
among answers 1,2,3 there is no correct one
A3. A swimmer swims against the current of the river. The speed of the river is 0.5 m/s, the speed of the swimmer relative to the water is 1.5 m/s. The speed modulus of the swimmer relative to the shore is equal to
1) 2 m/s 2) 1.5 m/s 3) 1 m/s 4) 0.5 m/s
A4. Moving in a straight line, one body covers a distance of 5 m every second. Another body, moving in a straight line in one direction, covers a distance of 10 m every second. The movements of these bodies
A5. The graph shows the dependence of the X coordinate of a body moving along the OX axis on time. What is the initial coordinate of the body?
3) -1 m 4) - 2 m
A6. What function v(t) describes the dependence of the velocity modulus on time for uniform rectilinear motion? (length is measured in meters, time in seconds)
1) v = 5t 2) v = 5/t 3) v = 5 4) v = -5
A7. The modulus of the body's velocity has doubled over some time. Which statement would be correct?
body acceleration doubled
acceleration decreased by 2 times
acceleration hasn't changed
body moves with acceleration
1) 1m/s 2 2) 1.2m/s 2 3) 2.0m/s 2 4) 2.4m/s 2
A9. When a body is in free fall, its speed (take g=10m/s 2)
in the first second it increases by 5 m/s, in the second – by 10 m/s;
in the first second it increases by 10 m/s, in the second – by 20 m/s;
in the first second it increases by 10 m/s, in the second – by 10 m/s;
in the first second it increases by 10m/s, and in the second – by 0m/s.
1) increased by 2 times 2) increased by 4 times
3) decreased by 2 times 4) decreased by 4 times
Option 2
A1. Two problems are solved:
A. the docking maneuver of two spacecraft is calculated;
b. the orbital period of spacecraft is calculated
around the Earth.
In what case can spaceships be considered as material points?
only in the first case
only in the second case
in both cases
neither in the first nor in the second case
1) 0 km 2) 109 km 3) 218 km 4) 436 km
A3. When they say that the change of day and night on Earth is explained by the rising and setting of the Sun, they mean a reference system associated
1) with the Sun 2) with the Earth
3) with the center of the galaxy 4) with any body
A4. When measuring the characteristics of the rectilinear movements of two material points, the values of the coordinates of the first point and the speed of the second point were recorded at the moments of time indicated in Tables 1 and 2, respectively:
What can be said about the nature of these movements, assuming that he hasn't changed in the time intervals between the moments of measurements?
1) both are uniform
2) the first is uneven, the second is uniform
3) the first is uniform, the second is uneven
4)both are uneven
A5. Using the graph of the distance traveled versus time, determine the speed
cyclist at time t = 2 s.
1) 2 m/s 2) 3 m/s
3) 6 m/s 4) 18 m/s
A6. The figure shows graphs of the distance traveled in one direction versus time for three bodies. Which body was moving with greater speed?
1) 1 2) 2 3) 3 4) the speeds of all bodies are the same 
A7. The speed of a body moving rectilinearly and uniformly accelerated changed when moving from point 1 to point 2 as shown in the figure. What direction does the acceleration vector have in this section?
A8. Using the graph of the velocity modulus versus time shown in the figure, determine the acceleration of a rectilinearly moving body at the time t=2s.
1) 2 m/s 2 2) 3 m/s 2 3) 9 m/s 2 4) 27 m/s 2
A9. In a tube from which the air has been evacuated, a pellet, a cork and a bird feather are simultaneously dropped from the same height. Which body will reach the bottom of the tube faster?
1) pellet 2) cork 3) bird feather 4) all three bodies at the same time.
A10. A car on a turn moves along a circular path of radius 50 m with a constant absolute speed of 10 m/s. What is the acceleration of the car?
1) 1 m/s 2 2) 2 m/s 2 3) 5 m/s 2 4) 0 m/s 2
Answers.
| Job number | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
| Option 1 | 3 | 4 | 3 | 1 | 3 | 3 | 4 | 1 | 3 | 2 |
| Option2 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 |
Profile level
Option 1
A1. A body thrown vertically upward reached a maximum height of 10 m and fell to the ground. The displacement module is equal to
1) 20m 2) 10m 3) 5m 4) 0m
A2. A body thrown vertically upward reached a maximum height of 5 m and fell to the ground. The distance traveled by the body is
1) 2.5m 2) 10m 3) 5m 4) 0m
A3. Two cars are moving along a straight highway: the first at a speed V, the second at a speed 4 V. What is the speed of the first car relative to the second?
1) 5V 2) 3V 3) -3V 4) -5V
A4. A small object comes off at point A from an airplane flying horizontally at speed V. What line is the trajectory of this object in the reference frame associated with the airplane, if air resistance is neglected?
A5. Two material points move along the OX axis according to the laws:
x 1 = 5 + 5t, x 2 = 5 - 5t (x - in meters, t - in seconds). What is the distance between them after 2 s?
1) 5m 2) 10m 3) 15m 4) 20m
A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5 + 15t 2 (X is measured in meters, time in seconds). The initial velocity module is equal to
A7. Two material points move in circles of radii R, = R and R 2 = 2R with the same speeds. Compare their centripetal accelerations.
1) a 1 = a 2 2)a 1 =2a 2 3)a 1 =a 2 /2 4)a 1 =4a 2
Part 2.
IN 1. The graph shows the dependence of movement speed on time. What is the average speed during the first five seconds?
AT 2. A small stone thrown from a flat horizontal surface of the earth at an angle to the horizon reached a maximum height of 4.05 m. How much time passed from the throw to the moment when its speed became directed horizontally?
Part 3.
C1. The coordinates of a moving body change according to the law X=3t+2, Y=-3+7t 2. Find the speed of the body 0.5 s after the start of movement.
Option 2
A1. A ball thrown vertically down from a height of 3 m bounces off the floor vertically and rises to a height of 3 m. The path of the ball is
1) -6m 2) 0m 3) 3m 4) 6m
A2. A stone thrown from a second floor window from a height of 4 m falls to the ground at a distance of 3 m from the wall of the house. What is the modulus of movement of the stone?
1) 3m 2) 4m 3) 5m 4) 7m
A3. A raft floats uniformly down the river at a speed of 6 km/h. A person moves across a raft at a speed of 8 km/h. What is the speed of a person in the reference frame associated with the shore?
1) 2 km/h 2) 7 km/h 3) 10 km/h 4) 14 km/h
A4. The helicopter rises vertically upward evenly. What is the trajectory of a point at the end of a helicopter rotor blade in the reference frame associated with the helicopter body?
3) point 4) helix
A5. A material point moves in a plane uniformly and rectilinearly according to the law: X = 4 + 3t, Y = 3 - 4t, where X,Y are the coordinates of the body, m; t - time, s. What is the speed of the body?
1) 1m/s 2) 3 m/s 3) 5 m/s 4) 7 m/s
A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5t+ 15t 2 (X is measured in meters, time in seconds).
The initial velocity module is equal to
1)0m/s 2) 5m/s 3) 7.5m/s 4) 15m/s
A7. The period of uniform motion of a material point along a circle is 2 s. After what minimum time does the direction of velocity change to the opposite?
1) 0.5 s 2) 1 s 3) 1.5 s 4) 2 s
Part 2.
IN 1. The graph shows the dependence of the speed V of the body on time t, describing the movement of the body along the OX axis. Determine the module of the average speed of movement in 2 seconds.
AT 2. A small stone was thrown from a flat horizontal surface of the earth at an angle to the horizon. What is the range of the stone if, 2 s after the throw, its speed was directed horizontally and equal to 5 m/s?
Part 3.
C1. A body emerging from a certain point moved with acceleration constant in magnitude and direction. Its speed at the end of the fourth second was 1.2 m/s, at the end of 7 seconds the body stopped. Find the path traveled by the body.
Answers.
| Job number | A1 | A2 | A3 | A4 | A5 | A6 | A7 | IN 1 | AT 2 | C1 |
| Option 1 | 4 | 2 | 3 | 3 | 4 | 1 | 2 | 1,6 | 0,9 | 7,6 |
| Option2 | 4 | 3 | 3 | 1 | 3 | 2 | 2 | 0,75 | 20 | 4,2 |
Test on the topic “Newton’s Laws. Forces in mechanics."
A basic level of
Option 1
A1. Which equality correctly expresses Hooke's law for an elastic spring?
1) F=kx 2) F x =kx 3) F x =-kx 4) F x =k | x |
A2. Which of the following bodies are associated with reference systems that cannot be considered inertial?
A . A skydiver descending at a steady speed.
B. A stone thrown vertically upward.
B. A satellite moving in orbit with a constant absolute velocity.
1) A 2) B 3) C 4) B and C
A3. Weight has a dimension
1) mass 2) acceleration 3) force 4) speed
A4. A body near the Earth's surface is in a state of weightlessness if it moves with an acceleration equal to the acceleration of gravity and directed
1) vertically down 2) vertically up
3) horizontally 4) at an acute angle to the horizontal.
A5. How will the sliding friction force change when the block moves along a horizontal plane if the normal pressure force is doubled?
1) will not change 2) will increase by 2 times
3) will decrease by 2 times 4) will increase by 4 times.
A6. What is the correct relationship between static friction force, sliding friction force and rolling friction force?
1) F tr.p =F tr >F tr.k 2) F tr.p >F tr >F tr.k 3) F tr.p F tr.k 4) F tr.p >F tr =F tr. .To
A7. A paratrooper launches uniformly at a speed of 6 m/s. The force of gravity acting on it is 800N. What is the mass of the skydiver?
1) 0 2) 60 kg 3) 80 kg 4) 140 kg.
A8. What is the measure of interaction between bodies?
1) Acceleration 2) Mass 3) Impulse. 4) Strength.
A9. How are changes in speed and inertia of a body related?
A . If the body is more inert, then the change in speed is greater.
B. If the body is more inert, then the change in speed is less.
B. A body that changes its speed faster is less inert.
G . The more inert body is the one that changes its speed faster.
1) A and B 2) B and D 3) A and D 4) B and C.
Option 2
A1. Which of the following formulas expresses the law of universal gravitation?
1) F=ma 2) F=μN 3) F x =-kx 4) F=Gm 1 m 2 /R 2
A2. When two cars collided, the buffer springs with a stiffness of 10 5 N/m were compressed by 10 cm. What is the maximum elastic force with which the springs acted on the car?
1) 10 4 N 2) 2*10 4 N 3) 10 6 N4) 2*10 6 N
A3. A body of mass 100 g lies on a horizontal stationary surface. Body weight is approximately
1) 0H 2) 1H 3) 100N 4) 1000 N.
A4. What is inertia?
2) the phenomenon of conservation of the speed of a body in the absence of the action of other bodies on it
3) change in speed under the influence of other bodies
4) movement without stopping.
A5. What is the dimension of the friction coefficient?
1) N/kg 2) kg/N 3) no dimension 4) N/s
A7. The student jumped to a certain height and sank to the ground. On what part of the trajectory did he experience the state of weightlessness?
1) when moving up 2) when moving down
3) only at the moment of reaching the top point 4) during the entire flight.
A8. What characteristics determine strength?
A. Module.
B. Direction.
B. Application point.
1) A, B, D 2) B and D 3) B, C, D 4) A, B, C.
A9. Which of the quantities (speed, force, acceleration, displacement) during mechanical motion always coincide in direction?
1) force and acceleration 2) force and speed
3) force and displacement 4) acceleration and displacement.
Answers.
| Job number | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 |
| Option 1 | 3 | 4 | 3 | 1 | 2 | 2 | 3 | 4 | 4 |
| Option2 | 4 | 1 | 2 | 2 | 3 | 1 | 4 | 4 | 1 |
Profile level
Option 1
A1. What forces in mechanics retain their significance during the transition from one inertial system to another?
1) forces of gravity, friction, elasticity.
2) only gravity
3) only friction force
4) only elastic force.
A2. How will the maximum static friction force change if the force of normal pressure of the block on the surface is doubled?
1) Will not change. 2) Will decrease by 2 times.
3) Will increase by 2 times. 4) Will increase 4 times.
A3. A block of mass 200 g slides on ice. Determine the sliding friction force acting on the block if the coefficient of sliding friction of the block on ice is 0.1.
1) 0.2N. 2) 2H. 3) 4H. 4) 20N
A4. How and how many times do you need to change the distance between the bodies so that the gravitational force decreases by 4 times?
1) Increase 2 times. 2) Reduce by 2 times.
3) Increase by 4 times. 4) Reduce by 4 times
A5. A load of mass m lies on the floor of an elevator starting to move downward with acceleration g.
What is the weight of this load?
1) mg. 2) m (g+a). 3) m (g-a). 4) 0
A6. After the rocket engines are turned off, the spacecraft moves vertically upward, reaches the top of the trajectory and then descends. At what part of the trajectory is the astronaut in a state of weightlessness? Neglect air resistance.
1) Only during upward movement. 2) Only during downward movement.
3) During the entire flight with the engine not running.
4) During the entire flight with the engine running.
Ticket 1.
Kinematics. Mechanical movement. Material point and absolutely rigid body. Kinematics of a material point and translational motion of a rigid body. Trajectory, path, displacement, speed, acceleration.
Ticket 2.
Kinematics of a material point. Speed, acceleration. Tangential, normal and total acceleration.
Kinematics- a branch of physics that studies the movement of bodies without being interested in the reasons that determine this movement.
Mechanicś logical movement́ nie - this is a change in body position in space relative to other bodies over time. (mechanical motion is characterized by three physical quantities: displacement, speed and acceleration)
The characteristics of mechanical motion are interconnected by basic kinematic equations:
Material point- a body whose dimensions, in the conditions of this problem, can be neglected.
Absolutely rigid body- a body whose deformation can be neglected under the conditions of a given problem.
Kinematics of a material point and translational motion of a rigid body: ?
movement in a rectangular, curvilinear coordinate system
how to write in different coordinate systems using a radius vector
Trajectory - some line described by the movement of the mat. points.
Path - scalar quantity characterizing the length of the trajectory of the body.
Moving - a straight line segment drawn from the initial position of a moving point to its final position (vector quantity)
Speed:
A vector quantity that characterizes the speed of movement of a particle along the trajectory in which this particle moves at each moment of time.
Derivative of the particle vector radius with respect to time.
Derivative of displacement with respect to time.
Acceleration:
A vector quantity characterizing the rate of change of the velocity vector.
Derivative of speed with respect to time.
Tangential acceleration - directed tangentially to the trajectory. Is a component of the acceleration vector a. Characterizes the change in speed modulo.
Centripetal or Normal acceleration - occurs when a point moves in a circle. Is a component of the acceleration vector a. The normal acceleration vector is always directed towards the center of the circle.
The total acceleration is the square root of the sum of the squares of the normal and tangential accelerations.
Ticket 3
Kinematics of rotational motion of a material point. Angular values. Relationship between angular and linear quantities.
Kinematics of rotational motion of a material point.
Rotational movement is a movement in which all points of the body describe circles, the centers of which lie on the same straight line, called the axis of rotation.
The axis of rotation passes through the center of the body, through the body, or may be located outside it.
Rotational motion of a material point is the movement of a material point in a circle.
Main characteristics of the kinematics of rotational motion: angular velocity, angular acceleration.
Angular displacement is a vector quantity that characterizes the change in angular coordinates during its movement.
Angular velocity is the ratio of the angle of rotation of the radius vector of a point to the period of time during which this rotation occurred. (direction along the axis around which the body rotates)
Rotation frequency is a physical quantity measured by the number of full revolutions made by a point per unit time with uniform movement in one direction (n)
Rotation period is the period of time during which a point makes a full revolution,
moving in a circle (T)
N is the number of revolutions made by the body during time t.
Angular acceleration is a quantity characterizing the change in the angular velocity vector over time.
Relationship between angular and linear quantities:
Relationship between linear and angular speed.
Relationship between tangential and angular acceleration.
the relationship between normal (centripetal) acceleration, angular velocity and linear velocity.
Ticket 4.
Dynamics of a material point. Classical mechanics, the limits of its applicability. Newton's laws. Inertial reference systems.
Dynamics of a material point:
Newton's laws
Laws of conservation (momentum, angular momentum, energy)
Classical mechanics is a branch of physics that studies the laws of changes in the positions of bodies and the causes that cause them, based on Newton's laws and Galileo's principle of relativity.
Classical mechanics is divided into:
statics (which considers the balance of bodies)
kinematics (which studies the geometric property of motion without considering its causes)
dynamics (which considers the movement of bodies).
Limits of applicability of classical mechanics:
At speeds close to the speed of light, classical mechanics stops working
The properties of the microcosm (atoms and subatomic particles) cannot be understood within the framework of classical mechanics
Classical mechanics becomes ineffective when considering systems with very large numbers of particles
Newton's first law (law of inertia):
There are reference systems relative to which a material point, in the absence of external influences, is at rest or moves uniformly and rectilinearly.
Newton's second law:
In an inertial reference frame, the product of the mass of a body and its acceleration is equal to the force acting on the body.
Newton's third law:
The forces with which interacting bodies act on each other are equal in magnitude and opposite in direction.
A reference system is a set of bodies that are not elevated relative to each other, in relation to which movements are considered (includes a reference body, a coordinate system, a clock)
An inertial reference system is a reference system in which the law of inertia is valid: any body that is not acted upon by external forces or the action of these forces is compensated is in a state of rest or uniform linear motion.
Inertia is a property inherent in bodies (it takes time to change the speed of a body).
Mass is a quantitative characteristic of inertia.
Ticket 5.
Center of mass (inertia) of the body. Momentum of a material point and a rigid body. Law of conservation of momentum. Movement of the center of mass.
The center of mass of a system of material points is a point whose position characterizes the distribution of the mass of the system in space.
distribution of masses in the coordinate system.
The position of the center of mass of a body depends on how its mass is distributed throughout the volume of the body.
The movement of the center of mass is determined only by external forces acting on the system. Internal forces of the system do not affect the position of the center of mass.
position of the center of mass.
The center of mass of a closed system moves in a straight line and uniformly or remains stationary.
The momentum of a material point is a vector quantity equal to the product of the mass of the point and its speed.
The momentum of a body is equal to the sum of the impulses of its individual elements.
Change in momentum mat. point is proportional to the applied force and has the same direction as the force.
Impulse of the mat system. points can only be changed by external forces, and the change in the momentum of the system is proportional to the sum of the external forces and coincides with it in direction. Internal forces, changing the impulses of individual bodies of the system, do not change the total impulse of the system.
Law of conservation of momentum:
if the sum of external forces acting on the body of the system is equal to zero, then the momentum of the system is conserved.
Ticket 6.
Work of force. Energy. Power. Kinetic and potential energy.Forces in nature.
Work is a physical quantity that characterizes the result of the action of a force and is numerically equal to the scalar product of the force vector and the displacement vector, completely under the influence of this force.
A = F S cosа (a-angle between the direction of force and the direction of movement)
No work is done if:
The force acts, but the body does not move
The body moves but the force is zero
The angle m/d by the force and displacement vectors is 90 degrees
Power is a physical quantity that characterizes the speed of work and is numerically equal to the ratio of work to the interval during which the work is performed.
Average power; instant power.
Power shows how much work is done per unit of time.
Energy is a scalar physical quantity, which is a single measure of various forms of motion of matter and a measure of the transition of the motion of matter from one form to another.
Mechanical energy is a quantity that characterizes the movement and interaction of bodies and is a function of the speeds and relative positions of bodies. It is equal to the sum of kinetic and potential energies.
A physical quantity equal to half the product of the mass of a body by the square of its speed is called the kinetic energy of the body.
Kinetic energy is the energy of motion.
A physical quantity equal to the product of the mass of a body by the acceleration modulus of gravity and the height to which the body is raised above the Earth’s surface is called the potential energy of interaction between the body and the Earth.
Potential energy is the energy of interaction.
A= – (Er2 – Er1).
1.Friction force.
Friction is one of the types of interaction between bodies. It occurs when two bodies come into contact. They arise due to the interaction between atoms and molecules of the bodies in contact. (Dry friction forces are the forces that arise when two solid bodies come into contact in the absence of a liquid or gaseous layer between them. The static friction force is always equal in magnitude to the external force and directed in the opposite direction. If the external force is greater than (Ftr)max, sliding friction occurs.)
μ is called the sliding friction coefficient.
2.Elasticity force. Hooke's law.
When a body is deformed, a force arises that strives to restore the previous size and shape of the body - the force of simplification.
(proportional to the deformation of the body and directed in the direction opposite to the direction of movement of body particles during deformation)
Fcontrol = –kx.
The coefficient k is called the rigidity of the body.
Tensile (x > 0) and compressive (x< 0).
Hooke's law: relative strain ε is proportional to stress σ, where E is Young's modulus.
3. Ground reaction force.
The elastic force acting on the body from the side of the support (or suspension) is called the support reaction force. When bodies come into contact, the support reaction force is directed perpendicular to the contact surface.
The weight of a body is the force with which the body, due to its attraction to the Earth, acts on a support or suspension.
4.Gravity. 
One of the manifestations of the force of universal gravity is the force of gravity. 
5.Gravitational force (gravitational force)
All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them.
Ticket 7.
Conservative and dissipative forces. Law of conservation of mechanical energy. Equilibrium condition for a mechanical system.
Conservative forces (potential forces) - forces whose work does not depend on the shape of the trajectory (depends only on the starting and ending points of application of forces)
Conservative forces are those forces whose work along any closed trajectory is equal to 0.
The work done by conservative forces along an arbitrary closed contour is 0;
A force acting on a material point is called conservative or potential if the work done by this force when moving this point from an arbitrary position 1 to another 2 does not depend on the trajectory along which this movement occurred:
Changing the direction of movement of a point along a trajectory to the opposite causes a change in the sign of the conservative force, since the quantity changes sign. Therefore, when a material point moves along a closed trajectory, for example, the work done by the conservative force is zero. 
Examples of conservative forces are the forces of universal gravitation, the force of elasticity, and the force of electrostatic interaction of charged bodies. A field whose work of forces in moving a material point along an arbitrary closed trajectory is zero is called potential.
Dissipative forces are forces, under the action of which on a moving mechanical system, its total mechanical energy decreases, turning into other, non-mechanical forms of energy, for example into heat.
example of dissipative forces: the force of viscous or dry friction.
Law of conservation of mechanical energy:
The sum of kinetic and potential energy of bodies that make up a closed system and interact with each other through gravitational and elastic forces remains unchanged.
Ek1 + Ep1 = Ek2 + Ep2
A closed system is a system that is not affected by external forces or is compensated for.
Equilibrium condition for a mechanical system:
Statics is a branch of mechanics that studies the conditions of equilibrium of bodies.
For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero.
If a body can rotate about a certain axis, then for its equilibrium it is not enough for the resultant of all forces to be zero.
Rule of moments: a body having a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body relative to this axis is equal to zero: M1 + M2 + ... = 0.
The length of the perpendicular drawn from the axis of rotation to the line of action of the force is called the arm of the force.
The product of the force modulus F and the arm d is called the moment of force M. The moments of those forces that tend to rotate the body counterclockwise are considered positive.
Ticket 8.
Kinematics of rotational motion of a rigid body. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular characteristics. Kinetic energy of rotational motion.
For a kinematic description of the rotation of a rigid body, it is convenient to use angular quantities: angular displacement Δφ, angular velocity ω
In these formulas, angles are expressed in radians. When a rigid body rotates relative to a fixed axis, all its points move with the same angular velocities and the same angular accelerations. The positive direction of rotation is usually taken to be counterclockwise.
Rotational motion of a rigid body:
1) around an axis - movement in which all points of the body lying on the axis of rotation are motionless, and the remaining points of the body describe circles with centers on the axis;
2) around a point - the movement of a body in which one of its points O is stationary, and all others move along the surfaces of spheres with a center at point O.
Kinetic energy of rotational motion.
Kinetic energy of rotational motion is the energy of a body associated with its rotation.
Let us divide the rotating body into small elements Δmi. Let us denote the distances to the axis of rotation by ri, and the linear velocity modules by υi. Then the kinetic energy of the rotating body can be written as:
The physical quantity depends on the distribution of masses of the rotating body relative to the axis of rotation. It is called the moment of inertia I of the body relative to a given axis:
In the limit as Δm → 0, this sum goes into an integral.
Thus, the kinetic energy of a rigid body rotating about a fixed axis can be represented as:
The kinetic energy of rotational motion is determined by the moment of inertia of the body relative to the axis of rotation and its angular velocity.
Ticket 9.
Dynamics of rotational motion. Moment of power. Moment of inertia. Steiner's theorem.
The moment of force is a quantity that characterizes the rotational effect of a force when it acts on a solid body. A distinction is made between the moment of force relative to the center (point) and relative to the axis.
1. The moment of force relative to the center O is a vector quantity. Its modulus Mo = Fh, where F is the modulus of the force, and h is the arm (the length of the perpendicular lowered from O to the line of action of the force)
Using the vector product, the moment of force is expressed by the equality Mo =, where r is the radius vector drawn from O to the point of application of the force.
2. The moment of force relative to an axis is an algebraic quantity equal to the projection onto this axis.
Moment of force (torque; rotational moment; torque) is a vector physical quantity equal to the product of the radius vector drawn from the axis of rotation to the point of application of the force and the vector of this force.
this expression is Newton's second law for rotational motion.
It is only true then:
a) if by moment M we mean part of the moment of an external force, under the influence of which the body rotates around an axis - this is the tangential component.
b) the normal component of the moment of force does not participate in the rotational motion, since Mn tries to displace the point from the trajectory, and by definition is identically equal to 0, with r- const Mn=0, and Mz determines the pressure force on the bearings.
The moment of inertia is a scalar physical quantity, a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion.
The moment of inertia depends on the mass of the body and on the location of the particles of the body relative to the axis of rotation.
Thin hoop Rod (fixed in the middle) Rod See
Homogeneous cylinder Disc Ball. 
(on the right is the picture for point 2 in Steiner’s volume)
Steiner's theorem.
The moment of inertia of a given body relative to any given axis depends not only on the mass, shape and size of the body, but also on the position of the body relative to this axis.
According to the Huygens-Steiner theorem, the moment of inertia of a body J relative to an arbitrary axis is equal to the sum:
1) the moment of inertia of this body Jо, relative to the axis passing through the center of mass of this body, and parallel to the axis under consideration,
2) the product of body mass by the square of the distance between the axes.
Ticket 10.
Moment of impulse. Basic equation for the dynamics of rotational motion (equation of moments). Law of conservation of angular momentum.
Momentum is a physical quantity that depends on how much mass is rotating and how it is distributed relative to the axis of rotation and at what speed the rotation occurs.
The angular momentum relative to a point is a pseudovector.
Momentum about an axis is a scalar quantity.
The angular momentum L of a particle relative to a certain reference point is determined by the vector product of its radius vector and momentum: L=
r is the radius vector of the particle relative to the selected reference point that is stationary in a given reference frame.
P is the momentum of the particle.
L = rp sin A = p l;
For systems rotating around one of the axes of symmetry (generally speaking, around the so-called principal axes of inertia), the following relation is valid:
moment of momentum of a body relative to the axis of rotation.
The angular momentum of a rigid body relative to the axis is the sum of the angular momentum of the individual parts.
Equation of moments.
The time derivative of the angular momentum of a material point relative to a fixed axis is equal to the moment of force acting on the point relative to the same axis:
M=JE=J dw/dt=dL/dt
The law of conservation of angular momentum (the law of conservation of angular momentum) - the vector sum of all angular momentum relative to any axis for a closed system remains constant in the case of equilibrium of the system. In accordance with this, the angular momentum of a closed system relative to any fixed point does not change with time.
=> dL/dt=0 i.e. L=const
Work and kinetic energy during rotational motion. Kinetic energy in plane motion.
External force applied to a point of mass
The distance traveled by the mass in time dt
But is equal to the modulus of the moment of force relative to the axis of rotation.
hence
taking into account that
we get the expression for work:
The work of rotational motion is equal to the work expended on turning the entire body.
Work during rotational motion occurs by increasing kinetic energy:
Plane (plane-parallel) motion is a motion in which all its points move parallel to some fixed plane.
Kinetic energy during plane motion is equal to the sum of the kinetic energies of translational and rotational motion:
Ticket 12.
Harmonic vibrations. Free undamped oscillations. Harmonic oscillator. Differential equation of a harmonic oscillator and its solution. Characteristics of undamped oscillations. Velocity and acceleration in undamped oscillations.
Mechanical vibrations are movements of bodies that repeat exactly (or approximately) at equal intervals of time. The law of motion of a body oscillating is specified using a certain periodic function of time x = f (t).
Mechanical vibrations, like oscillatory processes of any other physical nature, can be free and forced.
Free vibrations are carried out under the influence of the internal forces of the system, after the system has been brought out of equilibrium. Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations. Oscillations that occur under the influence of external periodically changing forces are called forced.
Harmonic oscillation is a phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function.
Oscillations are called harmonic if the following conditions are met:
1) the pendulum oscillations continue indefinitely (since there are no irreversible energy transformations);
2) its maximum deviation to the right from the equilibrium position is equal to the maximum deviation to the left;
3) the time of deviation to the right is equal to the time of deviation to the left;
4) the nature of the movement to the right and to the left from the equilibrium position is the same.
X = Xm cos (ωt + φ0).
V= -A w o sin(w o + φ)=A w o cos(w o t+ φ+P/2)
a= -A w o *2 cos(w o t+ φ)= A w o *2 cos(w o t+ φ+P)
x – displacement of the body from the equilibrium position,
xm – amplitude of oscillations, i.e. maximum displacement from the equilibrium position,
ω – cyclic or circular vibration frequency,
t – time.
φ = ωt + φ0 is called the phase of the harmonic process
φ0 is called the initial phase.
The minimum time interval through which the movement of the body is repeated is called the period of oscillation T
The oscillation frequency f shows how many oscillations occur in 1 s.
Undamped oscillations are oscillations with a constant amplitude.
Damped oscillations are oscillations whose energy decreases over time.
Free undamped oscillations:
Let's consider the simplest mechanical oscillatory system - a pendulum in a non-viscous medium.
Let's write the equation of motion according to Newton's second law:
Let's write this equation in projections onto the x-axis. Let's represent the acceleration projection onto the x-axis as the second derivative of the x-coordinate with respect to time.
Let's denote k/m by w2, and give the equation the form:
Where 
The solution to our equation is a function of the form:
A harmonic oscillator is a system that, when displaced from an equilibrium position, experiences a restoring force F proportional to the displacement x (according to Hooke's law):
k is a positive constant describing the rigidity of the system.
1.If F is the only force acting on the system, then the system is called a simple or conservative harmonic oscillator.
2. If there is also a frictional force (damping) proportional to the speed of movement (viscous friction), then such a system is called a damped or dissipative oscillator.
Differential equation of a harmonic oscillator and its solution:
As a model of a conservative harmonic oscillator, we take a load of mass m attached to a spring with stiffness k. Let x be the displacement of the load relative to the equilibrium position. Then, according to Hooke's law, a restoring force will act on it:
Using Newton's second law, we write:
Denoting and replacing acceleration with the second derivative of the coordinate with respect to time, we write:
This differential equation describes the behavior of a conservative harmonic oscillator. The coefficient ω0 is called the cyclic frequency of the oscillator.
We will look for a solution to this equation in the form:
Here is the amplitude, is the oscillation frequency (not necessarily equal to the natural frequency), and is the initial phase.
Substitute into the differential equation.
The amplitude is reduced. This means that it can have any value (including zero - this means that the load is at rest in the equilibrium position). You can also reduce by sine, since equality must be true at any time t. And the condition for the oscillation frequency remains:
The negative frequency can be discarded, since the arbitrariness in the choice of this sign is covered by the arbitrariness of the choice of the initial phase.
The general solution to the equation is written as:
where the amplitude A and the initial phase are arbitrary constants.
Kinetic energy is written as:
and there is potential energy
Characteristics of continuous oscillations:
Amplitude does not change
Frequency depends on stiffness and mass (spring)
Continuous oscillation speed:
Acceleration of continuous oscillations:
Ticket 13.
Free damped oscillations. Differential equation and its solution. Decrement, logarithmic decrement, damping coefficient. Relaxation time.
Free damped oscillations
If the forces of resistance to motion and friction can be neglected, then when the system is removed from the equilibrium position, only the elastic force of the spring will act on the load.
Let us write the equation of motion of the load, compiled according to Newton’s 2nd law:
Let's project the equation of motion onto the X axis.
transform: 
because
this is a differential equation of free harmonic undamped oscillations.
The solution to the equation is:
Differential equation and its solution:
In any oscillatory system there are resistance forces, the action of which leads to a decrease in the energy of the system. If the loss of energy is not replenished by the work of external forces, the oscillations will die out.
The resistance force is proportional to the speed:
r is a constant value called the resistance coefficient. The minus sign is due to the fact that force and velocity have opposite directions.
The equation of Newton's second law in the presence of resistance forces has the form:
Using the notation , , we rewrite the equation of motion as follows:
This equation describes the damped oscillations of the system
The solution to the equation is:
The attenuation coefficient is a value inversely proportional to the time during which the amplitude decreased by e times.
The time after which the amplitude of oscillations decreases by a factor of e is called the damping time
During this time, the system oscillates.
The damping decrement, a quantitative characteristic of the speed of damping of oscillations, is the natural logarithm of the ratio of two subsequent maximum deviations of the oscillating value in the same direction.
The logarithmic attenuation decrement is the logarithm of the ratio of amplitudes at the moments of successive passages of an oscillating quantity through a maximum or minimum (the attenuation of oscillations is usually characterized by a logarithmic attenuation decrement):
It is related to the number of oscillations N by the relation:
Relaxation time is the time during which the amplitude of a damped oscillation decreases by a factor of e.
Ticket 14.
Forced vibrations. Complete differential equation of forced oscillations and its solution. Period and amplitude of forced oscillations.
Forced oscillations are oscillations that occur under the influence of external forces that change over time.
Newton's second law for the oscillator (pendulum) will be written as:
If 
and replace the acceleration with the second derivative of the coordinate with respect to time, we obtain the following differential equation:
General solution of the homogeneous equation:
where A,φ are arbitrary constants
Let's find a particular solution. Let's substitute a solution of the form: into the equation and get the value for the constant:
Then the final solution will be written as:
The nature of forced oscillations depends on the nature of the action of the external force, on its magnitude, direction, frequency of action and does not depend on the size and properties of the oscillating body.
Dependence of the amplitude of forced oscillations on the frequency of the external force.
Period and amplitude of forced oscillations:
The amplitude depends on the frequency of forced oscillations; if the frequency is equal to the resonant frequency, then the amplitude is maximum. It also depends on the attenuation coefficient; if it is equal to 0, then the amplitude is infinite.
The period is related to the frequency; forced oscillations can have any period.
Ticket 15.
Forced vibrations. Period and amplitude of forced oscillations. Oscillation frequency. Resonance, resonant frequency. Family of resonance curves.
Ticket 14.
When the frequency of the external force and the frequency of the body’s own vibrations coincide, the amplitude of the forced vibrations increases sharply. This phenomenon is called mechanical resonance.
Resonance is the phenomenon of a sharp increase in the amplitude of forced oscillations.
An increase in amplitude is only a consequence of resonance, and the reason is the coincidence of the external frequency with the internal frequency of the oscillatory system.
Resonant frequency - the frequency at which the amplitude is maximum (slightly less than the natural frequency)
The graph of the amplitude of forced oscillations versus the frequency of the driving force is called a resonance curve.
Depending on the damping coefficient, we obtain a family of resonance curves; the lower the coefficient, the smaller the curve, the larger and higher it is.
Ticket 16.
Addition of oscillations of one direction. Vector diagram. Beating.
The addition of several harmonic oscillations of the same direction and the same frequency becomes clear if the oscillations are depicted graphically as vectors on a plane. The diagram obtained in this way is called a vector diagram.
Consider the addition of two harmonic oscillations of the same direction and the same frequency:
Let's represent both vibrations using vectors A1 and A2. Using the rules of vector addition, we construct the resulting vector A; the projection of this vector onto the x-axis is equal to the sum of the projections of the vectors being added:
Therefore, vector A represents the resulting oscillation. This vector rotates with the same angular velocity as vectors A1 and A2, so the sum of x1 and x2 is a harmonic oscillation with the same frequency, amplitude and phase. Using the cosine theorem, we find that
Representing harmonic oscillations using vectors allows you to replace the addition of functions with the addition of vectors, which is much simpler.
Beats are oscillations with periodically changing amplitude, resulting from the superposition of two harmonic oscillations with slightly different, but similar frequencies.
Ticket 17.
Addition of mutually perpendicular vibrations. Relationship between angular velocity of rotational motion and cyclic frequency. Lissajous figures.
Addition of mutually perpendicular vibrations:
Oscillations in two mutually perpendicular directions occur independently of each other:
Here the natural frequencies of harmonic oscillations are equal:
Let's consider the trajectory of cargo movement:
during the transformations we get:
Thus, the load will make periodic movements along an elliptical path. The direction of movement along the trajectory and the orientation of the ellipse relative to the axes depend on the initial phase difference
If the frequencies of two mutually perpendicular oscillations do not coincide, but are multiples, then the trajectories of motion are closed curves called Lissajous figures. Note that the ratio of oscillation frequencies is equal to the ratio of the numbers of points of contact of the Lissajous figure to the sides of the rectangle in which it is inscribed.
Ticket 18.
Oscillations of a load on a spring. Mathematical and physical pendulum. Characteristics of vibrations.
In order for free vibrations to occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement.
F (t) = ma (t) = –m ω2 x (t)
Fpr = –kx Hooke’s law.
The circular frequency ω0 of free oscillations of a load on a spring is found from Newton’s second law:
The frequency ω0 is called the natural frequency of the oscillatory system.
Therefore, Newton’s second law for a load on a spring can be written as:
The solution to this equation is harmonic functions of the form:
x = xm cos (ωt + φ0).
If the load, which was in the equilibrium position, was given an initial speed with the help of a sharp push
A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point suspended on a weightless inextensible thread or on a weightless rod in a gravitational field. The period of small oscillations of a mathematical pendulum of length l in a gravitational field with free fall acceleration g is equal to
and depends little on the amplitude and mass of the pendulum.
A physical pendulum is an oscillator, which is a solid body that oscillates in a field of any forces relative to a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of action of the forces and not passing through the center of mass of this body
Ticket 19.
Wave process. Elastic waves. Longitudinal and transverse waves. Plane wave equation. Phase speed. Wave equation and its solution.
A wave is a phenomenon of disturbance of a physical quantity propagating in space over time.
Depending on the physical medium in which the waves propagate, there are:
Waves on the surface of a liquid;
Elastic waves (sound, seismic waves);
Body waves (propagating through the medium);
Electromagnetic waves (radio waves, light, x-rays);
Gravitational waves;
Waves in plasma.
In relation to the direction of vibration of the particles of the medium:
Longitudinal waves (compression waves, P-waves) - particles of the medium oscillate parallel (along) the direction of propagation of the wave (as, for example, in the case of sound propagation);
Transverse waves (shear waves, S-waves) - particles of the medium oscillate perpendicular to the direction of propagation of the wave (electromagnetic waves, waves on the separation surfaces of media);
Mixed waves.
According to the type of wave front (surface of equal phases):
Plane wave - phase planes are perpendicular to the direction of wave propagation and parallel to each other;
Spherical wave - the surface of the phases is a sphere;
Cylindrical wave - the surface of the phases resembles a cylinder.
Elastic waves (sound waves) are waves propagating in liquid, solid and gaseous media due to the action of elastic forces.
Transverse waves are waves propagating in a direction perpendicular to the plane in which the displacements and vibrational velocities of particles are oriented.
Longitudinal waves, waves whose direction of propagation coincides with the direction of displacement of particles of the medium.
Plane wave, a wave in which all points lying in any plane perpendicular to the direction of its propagation at each moment correspond to the same displacements and velocities of particles of the medium
Plane wave equation:
Phase velocity is the speed of movement of a point with a constant phase of oscillatory motion in space along a given direction.
The geometric location of the points to which the oscillations reach at time t is called the wave front.
The geometric location of points oscillating in the same phase is called a wave surface.
Wave equation and its solution:
The propagation of waves in a homogeneous isotropic medium is generally described by the wave equation - a partial differential equation.
Where 
The solution to the equation is the equation of any wave, which has the form:
Ticket 20.
Transfer of energy by a traveling wave. Vector Umov. Addition of waves. Superposition principle. Standing wave.
A wave is a change in the state of a medium, propagating in this medium and carrying energy with it. (a wave is a spatial alternation of maxima and minima of any physical quantity that changes over time, for example, the density of a substance, electric field strength, temperature)
A traveling wave is a wave disturbance that changes in time t and space z according to the expression:
where is the amplitude envelope of the wave, K is the wave number and is the oscillation phase. The phase speed of this wave is given by
where is the wavelength.
Energy transfer - the elastic medium in which the wave propagates has both the kinetic energy of the vibrational motion of particles and the potential energy caused by the deformation of the medium.
A traveling wave, when propagating in a medium, transfers energy (unlike a standing wave).
Standing wave - oscillations in distributed oscillatory systems with a characteristic arrangement of alternating maxima (antinodes) and minima (nodes) of amplitude. In practice, such a wave occurs when reflected from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the incident one. In this case, the frequency, phase and attenuation coefficient of the wave at the place of reflection are extremely important. Examples of a standing wave include vibrations of a string, vibrations of air in an organ pipe
The Umov (Umov-Poynting) vector is the vector of the energy flux density of the physical field; is numerically equal to the energy transferred per unit time through a unit area perpendicular to the direction of energy flow at a given point.
The principle of superposition is one of the most general laws in many branches of physics.
In its simplest formulation, the principle of superposition states: the result of the action of several external forces on a particle is simply the sum of the results of the action of each of the forces.
The principle of superposition can also take other formulations, which, we emphasize, are completely equivalent to the one given above:
The interaction between two particles does not change when a third particle is introduced, which also interacts with the first two.
The interaction energy of all particles in a many-particle system is simply the sum of the energies of pairwise interactions between all possible pairs of particles. There are no many-particle interactions in the system.
The equations describing the behavior of a many-particle system are linear in the number of particles.
Addition of waves - addition of oscillations at each point.
The addition of standing waves is the addition of two identical waves propagating in different directions.
Ticket 21.
Inertial and non-inertial reference systems. Galileo's principle of relativity.
Inertial- such reference systems in which the body, which is not acted upon by forces, or they are balanced, is at rest or moves uniformly and rectilinearly
Non-inertial reference frame- an arbitrary reference system that is not inertial. Examples of non-inertial reference systems: a system moving rectilinearly with constant acceleration, as well as a rotating system
The principle of relativity Galilee- a fundamental physical principle according to which all physical processes in inertial reference systems proceed in the same way, regardless of whether the system is stationary or in a state of uniform and rectilinear motion.
It follows that all laws of nature are the same in all inertial frames of reference.
Ticket 22.
Physical foundations of molecular kinetic theory. Basic gas laws. Equation of state of an ideal gas. Basic equation of molecular kinetic theory.
Molecular kinetic theory (abbreviated MKT) is a theory that considers the structure of matter, mainly gases, from the point of view of three main approximately correct provisions:
all bodies consist of particles whose size can be neglected: atoms, molecules and ions;
particles are in continuous chaotic motion (thermal);
particles interact with each other through absolutely elastic collisions.
The main evidence for these provisions was considered:
Diffusion
Brownian motion
Changes in aggregate states of matter
Clapeyron-Mendeleev equation - a formula establishing the relationship between pressure, molar volume and absolute temperature of an ideal gas.
PV = υRT υ = m/μ
The Boyle-Mariotte law states:
At constant temperature and mass of an ideal gas, the product of its pressure and volume is constant
pV= const,
Where p- gas pressure; V- gas volume
Gay Lussac - V / T= const
Charles - P / T= const
Boyle - Mariotta - PV= const
Avogadro's law is one of the important fundamental principles of chemistry, which states that “equal volumes of different gases, taken at the same temperature and pressure, contain the same number of molecules.”
Corollary from Avogadro's law: one mole of any gas under the same conditions occupies the same volume.
In particular, under normal conditions, i.e. at 0 ° C (273 K) and 101.3 kPa, the volume of 1 mole of gas is 22.4 l/mol. This volume is called the molar volume of gas V m
Dalton's laws:
Law on the total pressure of a mixture of gases - The pressure of a mixture of chemically non-interacting ideal gases is equal to the sum of the partial pressures
Ptot = P1 + P2 + … + Pn
Law on the solubility of gas mixture components - At a constant temperature, the solubility in a given liquid of each of the components of the gas mixture located above the liquid is proportional to their partial pressure
Both Dalton's laws are strictly satisfied for ideal gases. For real gases, these laws are applicable provided that their solubility is low and their behavior is close to that of an ideal gas.
Equation of states of an ideal gas - see Clapeyron - Mendeleev equation PV = υRT υ = m/μ
The basic equation of molecular kinetic theory (MKT) is
Basic equation of molecular kinetic theory. Gas pressure on the wall. Average energy of molecules. Law of equidistribution. Number of degrees of freedom.
Gas pressure on the wall - During their movement, molecules collide with each other, as well as with the walls of the vessel in which the gas is located. There are many molecules in a gas, so the number of their impacts is very large. Although the impact force of an individual molecule is small, the effect of all molecules on the walls of the vessel is significant, and it creates gas pressure
Average energy of a molecule –
The average kinetic energy of gas molecules (per one molecule) is determined by the expression
Ek= ½ m
The kinetic energy of the translational motion of atoms and molecules, averaged over a huge number of randomly moving particles, is a measure of what is called temperature. If the temperature T is measured in degrees Kelvin (K), then its relationship with E k is given by the relation
The law of equipartition is a law of classical statistical physics, which states that for a statistical system in a state of thermodynamic equilibrium, for each translational and rotational degree of freedom there is an average kinetic energy kT/2, and for each vibrational degree of freedom - the average energy kT(Where T - absolute temperature of the system, k - Boltzmann constant).
The equipartition theorem states that in thermal equilibrium, energy is divided equally between its different forms
The number of degrees of freedom is the smallest number of independent coordinates that determine the position and configuration of the molecule in space.
The number of degrees of freedom for a monatomic molecule is 3 (translational motion in the direction of three coordinate axes), for diatomic - 5 (three translational and two rotational, since rotation around the X axis is possible only at very high temperatures), for triatomic - 6 (three translational and three rotational).
Ticket 24.
Elements of classical statistics. Distribution functions. Maxwell distribution by absolute value of velocities.
Ticket 25.
Maxwell distribution by absolute value of velocity. Finding the characteristic velocities of molecules.
Elements of classical statistics:
A random variable is a quantity that, as a result of experiment, takes on one of many values, and the appearance of one or another value of this quantity cannot be accurately predicted before its measurement.
A continuous random variable (CRV) is a random variable that can take all values from some finite or infinite interval. The set of possible values of a continuous random variable is infinite and uncountable.
The distribution function is the function F(x), which determines the probability that the random variable X as a result of the test will take a value less than x.
Distribution function is the probability density of the distribution of particles of a macroscopic system over coordinates, momenta or quantum states. The distribution function is the main characteristic of a wide variety of (not only physical) systems that are characterized by random behavior, i.e. random change in the state of the system and, accordingly, its parameters.
Maxwell distribution by absolute value of velocities:
Gas molecules constantly collide as they move. The speed of each molecule upon collision changes. It can increase and decrease. However, the RMS speed remains unchanged. This is explained by the fact that in a gas at a certain temperature, a certain stationary velocity distribution of molecules that does not change over time is established, which obeys a certain statistical law. The speed of an individual molecule may change over time, but the proportion of molecules with speeds in a certain speed range remains unchanged.
Graph of the ratio of the fraction of molecules to the speed interval Δv i.e. .
In practice, the graph is described by the velocity distribution function of molecules or Maxwell’s law:
Derived formula:
When the gas temperature changes, the speed of movement of all molecules will change, and, consequently, the most probable speed. Therefore, the maximum of the curve will shift to the right as the temperature increases and to the left as the temperature decreases.
The height of the maximum changes with temperature changes. The fact that the distribution curve begins at the origin means that there are no stationary molecules in the gas. From the fact that the curve asymptotically approaches the x-axis at infinitely high speeds, it follows that there are few molecules with very high speeds.
Ticket 26.
Boltzmann distribution. Maxwell-Boltzmann distribution. Boltzmann's barometric formula.
Boltzmann distribution is the energy distribution of particles (atoms, molecules) of an ideal gas under conditions of thermodynamic equilibrium.
Boltzmann distribution law:
where n is the concentration of molecules at height h,
n0 – concentration of molecules at the initial level h = 0,
m – mass of particles,
g – free fall acceleration,
k – Boltzmann constant,
T – temperature.
Maxwell-Boltzmann distribution:
equilibrium distribution of ideal gas particles by energy (E) in an external force field (for example, in a gravitational field); determined by the distribution function:
where E is the sum of the kinetic and potential energies of the particle,
T - absolute temperature,
k - Boltzmann constant
The barometric formula is the dependence of the pressure or density of a gas on the height in the gravitational field. For an ideal gas that has a constant temperature T and is located in a uniform gravitational field (at all points of its volume the acceleration of gravity g is the same), the barometric formula has the following form:
where p is the gas pressure in the layer located at height h,
p0 - pressure at zero level (h = h0),
M is the molar mass of the gas,
R - gas constant,
T - absolute temperature.
From the barometric formula it follows that the concentration of molecules n (or gas density) decreases with altitude according to the same law:
where m is the mass of a gas molecule, k is Boltzmann’s constant.
Ticket 27.
The first law of thermodynamics. Work and warmth. Processes. Work done by gas in various isoprocesses. The first law of thermodynamics in various processes. Formulations of the first principle.
Ticket 28.
Internal energy of an ideal gas. Heat capacity of an ideal gas at constant volume and constant pressure. Mayer's equation.
The first law of thermodynamics - one of the three basic laws of thermodynamics, is the law of conservation of energy for thermodynamic systems
There are several equivalent formulations of the first law of thermodynamics:
1) The amount of heat received by the system goes to change its internal energy and perform work against external forces
2) The change in the internal energy of a system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system and does not depend on the method in which this transition is carried out
3) The change in the total energy of the system in a quasi-static process is equal to the amount of heat Q, communicated to the system, in sum with the change in energy associated with the amount of matter N at chemical potential μ, and work A"performed on the system by external forces and fields, minus the work A committed by the system itself against external forces
ΔU = Q - A + μΔΝ + A`
An ideal gas is a gas in which it is assumed that the potential energy of the molecules can be neglected compared to their kinetic energy. There are no forces of attraction or repulsion between molecules, collisions of particles with each other and with the walls of the vessel are absolutely elastic, and the interaction time between molecules is negligible compared to the average time between collisions.
Work - When expanding, the work of a gas is positive. When compressed, it is negative. Thus:
A" = pDV - gas work (A" - gas expansion work)
A= - pDV - work of external forces (A - work of external forces on gas compression)
Heat-kinetic part of the internal energy of a substance, determined by the intense chaotic movement of the molecules and atoms of which this substance consists.
The heat capacity of an ideal gas is the ratio of the heat imparted to the gas to the temperature change δT that occurred.
The internal energy of an ideal gas is a quantity that depends only on its temperature and does not depend on volume.
Mayer's equation shows that the difference in the heat capacities of a gas is equal to the work done by one mole of an ideal gas when its temperature changes by 1 K, and explains the meaning of the universal gas constant R.
For any ideal gas, Mayer's relation is valid:
, 
Processes:
An isobaric process is a thermodynamic process occurring in a system at constant pressure.
The work done by the gas during expansion or compression of the gas is equal to
Work done by gas during expansion or compression of gas:
The amount of heat received or given off by the gas:
at a constant temperature dU = 0, therefore the entire amount of heat imparted to the system is spent on doing work against external forces.
Heat capacity:
Ticket 29.
Adiabatic process. Adiabatic equation. Poisson's equation. Work in an adiabatic process.
An adiabatic process is a thermodynamic process in a macroscopic system in which the system neither receives nor releases thermal energy.
For an adiabatic process, the first law of thermodynamics, due to the absence of heat exchange between the system and the environment, has the form:
In an adiabatic process, heat exchange with the environment does not occur, i.e. δQ=0. Consequently, the heat capacity of an ideal gas in an adiabatic process is also zero: Sadiab=0.
Work is done by the gas due to changes in internal energy Q=0, A=-DU
In an adiabatic process, the gas pressure and its volume are related by the relation:
pV*g=const, where g= Cp/Cv.
In this case, the following relations are valid:
p2/p1=(V1/V2)*g, *g-degree
T2/T1=(V1/V2)*(g-1), *(g-1)-degree
T2/T1=(p2/p1)*(g-1)/g. *(g-1)/g -degree
The given relations are called Poisson’s equations
equation of the adiabatic process. (Poisson equation) g - adiabatic exponent
Ticket 30.
Second law of thermodynamics. Carnot cycle. Efficiency of an ideal heat engine. Entropy and thermodynamic probability. Various formulations of the second law of thermodynamics.
The second law of thermodynamics is a physical principle that imposes restrictions on the direction of heat transfer processes between bodies.
The second law of thermodynamics states that spontaneous transfer of heat from a less heated body to a more heated body is impossible.
The second law of thermodynamics prohibits the so-called perpetual motion machines of the second kind, showing the impossibility of converting all the internal energy of the system into useful work.
The second law of thermodynamics is a postulate that cannot be proven within the framework of thermodynamics. It was created on the basis of a generalization of experimental facts and received numerous experimental confirmations.
Clausius's postulate: “A process is impossible, the only result of which would be the transfer of heat from a colder body to a hotter one”(this process is called Clausius process).
Thomson's postulate: “A circular process is impossible, the only result of which would be the production of work by cooling the thermal reservoir”(this process is called Thomson process).
The Carnot cycle is an ideal thermodynamic cycle.
A Carnot heat engine operating in this cycle has the highest efficiency of all machines in which the maximum and minimum temperatures of the cycle being carried out coincide, respectively, with the maximum and minimum temperatures of the Carnot cycle.
The Carnot cycle consists of four stages:
1.Isothermal expansion (in the figure - process A→B). At the beginning of the process, the working fluid has a temperature Tn, that is, the temperature of the heater. The body is then brought into contact with a heater, which transfers an amount of heat QH to it isothermally (at a constant temperature). At the same time, the volume of the working fluid increases.
2. Adiabatic (isentropic) expansion (in the figure - process B→C). The working fluid is disconnected from the heater and continues to expand without heat exchange with the environment. At the same time, its temperature decreases to the temperature of the refrigerator.
3.Isothermal compression (in the figure - process B→G). The working fluid, which by that time has a temperature TX, is brought into contact with the refrigerator and begins to compress isothermally, giving the amount of heat QX to the refrigerator.
4. Adiabatic (isentropic) compression (in the figure - process G→A). The working fluid is disconnected from the refrigerator and compressed without heat exchange with the environment. At the same time, its temperature increases to the temperature of the heater.
Entropy- an indicator of randomness or disorder in the structure of a physical system. In thermodynamics, entropy expresses the amount of thermal energy available to do work: the less energy, the less entropy. On the scale of the Universe, entropy increases. Energy can be extracted from a system only by transforming it into a less ordered state. According to the second law of thermodynamics, entropy in an isolated system either does not increase or increases during any process.
Thermodynamic probability, the number of ways in which the state of a physical system can be realized. In thermodynamics, the state of a physical system is characterized by certain values of density, pressure, temperature and other measurable quantities.
Ticket 31.
Micro- and macrostates. Statistical weight. Reversible and irreversible processes. Entropy. Law of increasing entropy. Nernst's theorem.
Ticket 30.
Statistical weight is the number of ways in which a given system state can be realized. The statistical weights of all possible states of the system determine its entropy.
Reversible and irreversible processes.
A reversible process (that is, equilibrium) is a thermodynamic process that can occur in both the forward and reverse directions, passing through the same intermediate states, and the system returns to its original state without energy expenditure, and no macroscopic changes remain in the environment.
(A reversible process can be made to flow in the opposite direction at any time by changing any independent variable by an infinitesimal amount.
Reversible processes produce the most work.
In practice, a reversible process cannot be realized. It flows infinitely slowly, and you can only get closer to it.)
An irreversible process is a process that cannot be carried out in the opposite direction through all the same intermediate states. All real processes are irreversible.
In an adiabatically isolated thermodynamic system, entropy cannot decrease: it is either preserved if only reversible processes occur in the system, or increases if at least one irreversible process occurs in the system.
The written statement is another formulation of the second law of thermodynamics.
Nernst's theorem (Third law of thermodynamics) is a physical principle that determines the behavior of entropy as temperature approaches absolute zero. It is one of the postulates of thermodynamics, accepted on the basis of a generalization of a significant amount of experimental data.
The third law of thermodynamics can be formulated as follows:
“The increase in entropy at absolute zero temperature tends to a finite limit, independent of the equilibrium state the system is in.”
Where x is any thermodynamic parameter.
(The third law of thermodynamics applies only to equilibrium states.
Since, based on the second law of thermodynamics, entropy can only be determined up to an arbitrary additive constant (that is, it is not the entropy itself that is determined, but only its change):
The third law of thermodynamics can be used to accurately determine entropy. In this case, the entropy of the equilibrium system at absolute zero temperature is considered equal to zero.
According to the third law of thermodynamics, at value .)
Ticket 32.
Real gases. Van de Waals equation. The internal energy is really gas.
A real gas is a gas that is not described by the Clapeyron-Mendeleev equation of state for an ideal gas.
Molecules in a real gas interact with each other and occupy a certain volume.
In practice, it is often described by the generalized Mendeleev-Clapeyron equation:
The van der Waals gas equation of state is an equation that relates the basic thermodynamic quantities in the van der Waals gas model.
(To more accurately describe the behavior of real gases at low temperatures, a van der Waals gas model was created that takes into account the forces of intermolecular interaction. In this model, the internal energy U becomes a function of not only temperature, but also volume.)
The thermal equation of state (or, often, simply the equation of state) is the relationship between pressure, volume and temperature.
For n moles of van der Waals gas, the equation of state looks like this:
T - absolute temperature,
R is the universal gas constant.
p - pressure,
The internal energy of a real gas consists of the kinetic energy of the thermal motion of molecules and the potential energy of intermolecular interaction
Ticket 33.
Physical kinetics. The phenomenon of transport in gases. Number of collisions and mean free path of molecules.
Physical kinetics is a microscopic theory of processes in nonequilibrium media. In kinetics, the methods of quantum or classical statistical physics are used to study the processes of transfer of energy, momentum, charge and matter in various physical systems (gases, plasma, liquids, solids) and the influence of external fields on them.
Transport phenomena in gases are observed only if the system is in a nonequilibrium state.
Diffusion is the process of transferring matter or energy from an area of high concentration to an area of low concentration.
Thermal conductivity is the transfer of internal energy from one part of the body to another or from one body to another upon their direct contact.
Number (Frequency) of collisions and mean free path of molecules.
Moving at medium speed
< l > =
τ is the time that a molecule moves between two successive collisions (analogous to a period)
Then the average number of collisions per unit time (average collision frequency) is the reciprocal of the period:
v= 1 / τ =
Path length< l>, at which the probability of collision with target particles becomes equal to one, is called the mean free path.
Ticket 34.
Diffusion in gases. Diffusion coefficient. Viscosity of gases. Viscosity coefficient. Thermal conductivity. Coefficient of thermal conductivity.
Diffusion is the process of transferring matter or energy from an area of high concentration to an area of low concentration.
Diffusion in gases occurs much faster than in other states of aggregation, which is due to the nature of the thermal movement of particles in these media.
Diffusion coefficient - the amount of a substance passing per unit time through a section of unit area with a concentration gradient equal to unity.
The diffusion coefficient reflects the rate of diffusion and is determined by the properties of the medium and the type of diffusing particles.
Viscosity (internal friction) is one of the transfer phenomena, the property of fluid bodies (liquids and gases) to resist the movement of one part relative to another.
When talking about viscosity, the number that is usually considered is viscosity coefficient. There are several different viscosity coefficients, depending on the acting forces and the nature of the fluid:
Dynamic viscosity (or absolute viscosity) determines the behavior of an incompressible Newtonian fluid.
Kinematic viscosity is the dynamic viscosity divided by the density for Newtonian fluids.
Bulk viscosity determines the behavior of a compressible Newtonian fluid.
Shear Viscosity (Shear Viscosity) – coefficient of viscosity under shear loads (for non-Newtonian fluids)
Bulk viscosity - compression viscosity coefficient (for non-Newtonian fluids)
Thermal conduction is the process of heat transfer, leading to equalization of temperature throughout the entire volume of the system.
Thermal conductivity coefficient is a numerical characteristic of the thermal conductivity of a material, equal to the amount of heat passing through a material with a thickness of 1 m and an area of 1 sq.m per hour when the temperature difference on two opposite surfaces is 1 degree C.
A basic level of
Option 1
A1. The trajectory of a moving material point in a finite time is
line segment
part of the plane
finite set of points
among answers 1,2,3 there is no correct one
A2. The chair was moved first by 6 m, and then by another 8 m. What is the modulus of total displacement?
1) 2 m 2) 6 m 3) 10 m 4) cannot be determined
A3. A swimmer swims against the current of the river. The speed of the river is 0.5 m/s, the speed of the swimmer relative to the water is 1.5 m/s. The speed modulus of the swimmer relative to the shore is equal to
1) 2 m/s 2) 1.5 m/s 3) 1 m/s 4) 0.5 m/s
A4. Moving in a straight line, one body covers a distance of 5 m every second. Another body, moving in a straight line in one direction, covers a distance of 10 m every second. The movements of these bodies
A5. The graph shows the dependence of the coordinate X of a body moving along the OX axis on time. What is the initial coordinate of the body?
3) -1 m 4) - 2 m
A6. What function v(t) describes the dependence of the velocity modulus on time for uniform rectilinear motion? (length is measured in meters, time in seconds)
1) v= 5t2)v= 5/t3)v= 5 4)v= -5
A7. The modulus of the body's velocity has doubled over some time. Which statement would be correct?
body acceleration doubled
acceleration decreased by 2 times
acceleration hasn't changed
body moves with acceleration
A8. The body, moving rectilinearly and uniformly accelerated, increased its speed from 2 to 8 m/s in 6 s. What is the acceleration of the body?
1) 1m/s 2 2) 1.2m/s 2 3) 2.0m/s 2 4) 2.4m/s 2
A9. When a body is in free fall, its speed (take g = 10 m/s 2)
in the first second it increases by 5 m/s, in the second – by 10 m/s;
in the first second it increases by 10 m/s, in the second – by 20 m/s;
in the first second it increases by 10 m/s, in the second – by 10 m/s;
in the first second it increases by 10m/s, and in the second – by 0m/s.
A10. The speed of rotation of the body in a circle increased by 2 times. Centripetal acceleration of a body
1) increased by 2 times 2) increased by 4 times
3) decreased by 2 times 4) decreased by 4 times
Option 2
A1. Two problems are solved:
A. the docking maneuver of two spacecraft is calculated;
b. The period of revolution of spacecraft around the Earth is calculated.
In what case can spaceships be considered as material points?
only in the first case
only in the second case
in both cases
neither in the first nor in the second case
A2. The car drove around Moscow twice along the ring road, which is 109 km long. The distance traveled by the car is
1) 0 km 2) 109 km 3) 218 km 4) 436 km
A3. When they say that the change of day and night on Earth is explained by the rising and setting of the Sun, they mean a reference system associated
1) with the Sun 2) with the Earth
3) with the center of the galaxy 4) with any body
A4. When measuring the characteristics of the rectilinear movements of two material points, the values of the coordinates of the first point and the speed of the second point were recorded at the moments of time indicated in Tables 1 and 2, respectively:
What can be said about the nature of these movements, assuming that he hasn't changed in the time intervals between the moments of measurements?
1) both are uniform
2) the first is uneven, the second is uniform
3) the first is uniform, the second is uneven
4)both are uneven
A5. Using the graph of the distance traveled versus time, determine the speed of the cyclist at time t = 2 s. 1) 2 m/s 2) 3 m/s
3) 6 m/s4) 18 m/s
A6. The figure shows graphs of the distance traveled in one direction versus time for three bodies. Which body was moving with greater speed? 1) 1 2) 2 3) 34) the speeds of all bodies are the same
A7. The speed of a body moving rectilinearly and uniformly accelerated changed when moving from point 1 to point 2 as shown in the figure. What direction does the acceleration vector have in this section?
A8. Using the graph of the velocity modulus versus time shown in the figure, determine the acceleration of a rectilinearly moving body at time t=2s.
1) 2 m/s 2 2) 3 m/s 2 3) 9 m/s 2 4) 27 m/s 2
A9. In a tube from which the air has been evacuated, a pellet, a cork and a bird feather are simultaneously dropped from the same height. Which body will reach the bottom of the tube faster?
1) pellet 2) cork 3) bird feather 4) all three bodies at the same time.
A10. A car on a turn moves along a circular path of radius 50 m with a constant absolute speed of 10 m/s. What is the acceleration of the car?
1) 1 m/s 2 2) 2 m/s 2 3) 5 m/s 2 4) 0 m/s 2
Answers.
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Basic concepts of kinematics and kinematic characteristics
Human movement is mechanical, that is, it is a change in the body or its parts relative to other bodies. Relative movement is described by kinematics.
Kinematics – a branch of mechanics in which mechanical motion is studied, but the causes of this motion are not considered. The description of the movement of both the human body (its parts) in various sports and various sports equipment is an integral part of sports biomechanics and in particular kinematics.
Whatever material object or phenomenon we consider, it turns out that nothing exists outside of space and outside of time. Any object has spatial dimensions and shape, and is located in some place in space in relation to another object. Any process in which material objects participate has a beginning and an end in time, how long it lasts in time, and can occur earlier or later than another process. This is precisely why there is a need to measure spatial and temporal extent.
Basic units of measurement of kinematic characteristics in the international system of measurements SI.
Space. One forty-millionth of the length of the earth's meridian passing through Paris was called a meter. Therefore, length is measured in meters (m) and its multiple units: kilometers (km), centimeters (cm), etc.
Time– one of the fundamental concepts. We can say that this is what separates two successive events. One way to measure time is to use any regularly repeated process. One eighty-six thousandth of an earthly day was chosen as a unit of time and was called the second (s) and its multiple units (minutes, hours, etc.).
In sports, special time characteristics are used:
Moment of time(t)- this is a temporary measure of the position of a material point, links of a body or system of bodies. Moments of time indicate the beginning and end of a movement or any part or phase of it.
Movement duration(∆t) – this is its temporary measure, which is measured by the difference between the moments of the end and the beginning of movement∆t = tcon. – tbeg.
Movement speed(N) – it is a temporal measure of the repetition of movements repeated per unit of time. N = 1/∆t; (1/s) or (cycle/s).
Rhythm of movements – this is a temporary measure of the relationship between parts (phases) of movements. It is determined by the ratio of the duration of the parts of the movement.
The position of a body in space is determined relative to a certain reference system, which includes a reference body (that is, relative to which the movement is considered) and a coordinate system necessary to describe at a qualitative level the position of the body in one or another part of space.
The beginning and direction of measurement are associated with the reference body. For example, in a number of competitions, the origin of coordinates can be chosen as the start position. Various competitive distances in all cyclic sports are already calculated from it. Thus, in the selected “start-finish” coordinate system, the distance in space that the athlete will move when moving is determined. Any intermediate position of the athlete’s body during movement is characterized by the current coordinate within the selected distance interval.
To accurately determine a sports result, the competition rules stipulate at what point (reference point) the count is taken: along the toe of a skater’s skate, at the protruding point of a sprinter’s chest, or along the back edge of the landing long jumper’s track.
In some cases, to accurately describe the movement of the laws of biomechanics, the concept of a material point is introduced.
Material point – this is a body whose dimensions and internal structure can be neglected under given conditions.
The movement of bodies can be different in nature and intensity. To characterize these differences, a number of terms are introduced in kinematics, presented below.
Trajectory – a line described in space by a moving point of a body. When biomechanical analysis of movements, first of all, the trajectories of movements of characteristic points of a person are considered. As a rule, such points are the joints of the body. Based on the type of movement trajectories, they are divided into rectilinear (straight line) and curvilinear (any line other than a straight line).
Moving – is the vector difference between the final and initial position of the body. Therefore, displacement characterizes the final result of the movement.
Path – this is the length of the trajectory section traversed by a body or a point of the body during a selected period of time.
KINEMATICS OF A POINT
Introduction to Kinematics
Kinematics is a branch of theoretical mechanics that studies the motion of material bodies from a geometric point of view, regardless of the applied forces.
The position of a moving body in space is always determined in relation to any other unchanging body, called reference body. A coordinate system invariably associated with a reference body is called reference system. In Newtonian mechanics, time is considered absolute and not related to moving matter. In accordance with this, it proceeds identically in all reference systems, regardless of their motion. The basic unit of time is the second (s).
If the position of the body relative to the chosen frame of reference does not change over time, then it is said that body relative to a given frame of reference is at rest. If a body changes its position relative to the chosen reference system, then it is said to move relative to this system. A body can be at rest in relation to one reference system, but move (and in completely different ways) in relation to other reference systems. For example, a passenger sitting motionless on the bench of a moving train is at rest relative to the frame of reference associated with the car, but is moving with respect to the frame of reference associated with the Earth. A point lying on the rolling surface of the wheel moves in relation to the reference system associated with the car in a circle, and in relation to the reference system associated with the Earth, in a cycloid; the same point is at rest with respect to the coordinate system associated with the wheel pair.
Thus, the movement or rest of a body can be considered only in relation to any chosen frame of reference. Set the motion of a body relative to some reference system -means to give functional dependencies with the help of which one can determine the position of the body at any time relative to this system. Different points of the same body move differently in relation to the chosen reference system. For example, in relation to the system associated with the Earth, the tread surface point of the wheel moves along a cycloid, and the center of the wheel moves in a straight line. Therefore, the study of kinematics begins with the kinematics of a point.
§ 2. Methods for specifying the movement of a point
The movement of a point can be specified in three ways:natural, vector and coordinate.
With the natural way The movement assignment is given by a trajectory, i.e., a line along which the point moves (Fig. 2.1). On this trajectory, a certain point is selected, taken as the origin. The positive and negative directions of reference of the arc coordinate, which determines the position of the point on the trajectory, are selected. As the point moves, the distance will change. Therefore, to determine the position of a point at any time, it is enough to specify the arc coordinate as a function of time:
This equality is called equation of motion of a point along a given trajectory .
So, the movement of a point in the case under consideration is determined by a combination of the following data: the trajectory of the point, the position of the origin of the arc coordinate, the positive and negative directions of the reference and the function .
With the vector method of specifying the movement of a point, the position of the point is determined by the magnitude and direction of the radius vector drawn from the fixed center to a given point (Fig. 2.2). When a point moves, its radius vector changes in magnitude and direction. Therefore, to determine the position of a point at any time, it is enough to specify its radius vector as a function of time:
This equality is called vector equation of motion of a point .
With the coordinate method
specifying the motion, the position of the point in relation to the selected reference system is determined using a rectangular Cartesian coordinate system (Fig. 2.3). When a point moves, its coordinates change over time. Therefore, to determine the position of a point at any time, it is enough to specify the coordinates , ,
as a function of time:
These equalities are called equations of motion of a point in rectangular Cartesian coordinates . The motion of a point in a plane is determined by two equations of system (2.3), rectilinear motion by one.
There is a mutual connection between the three described methods of specifying movement, which allows you to move from one method of specifying movement to another. This is easy to verify, for example, when considering the transition from the coordinate method of specifying movement to vector.
Let us assume that the motion of a point is given in the form of equations (2.3). Bearing in mind that
can be written down
And this is an equation of the form (2.2).
Task 2.1.
Find the equation of motion and the trajectory of the middle point of the connecting rod, as well as the equation of motion of the slider of the crank-slider mechanism (Fig. 2.4), if 
;
.
Solution. The position of a point is determined by two coordinates and . From Fig. 2.4 it is clear that
, .
Then from and:
; ; 
.
Substituting values , 
and , we obtain the equations of motion of the point:
; 
.
To find the equation for the trajectory of a point in explicit form, it is necessary to exclude time from the equations of motion. For this purpose, we will carry out the necessary transformations in the equations of motion obtained above:
; .
By squaring and adding the left and right sides of these equations, we obtain the trajectory equation in the form
.
Therefore, the trajectory of the point is an ellipse.
The slider moves in a straight line. The coordinate , which determines the position of the point, can be written in the form
.
Speed and acceleration
Point speed
In the previous article, the movement of a body or point is defined as a change in position in space over time. In order to more fully characterize the qualitative and quantitative aspects of movement, the concepts of speed and acceleration were introduced.
Velocity is a kinematic measure of the movement of a point, characterizing the speed of change of its position in space.
Velocity is a vector quantity, that is, it is characterized not only by its magnitude (scalar component), but also by its direction in space.
As is known from physics, with uniform motion, speed can be determined by the length of the path traveled per unit time: v = s/t = const
(it is assumed that the origin of the path and time are the same).
During rectilinear motion, the speed is constant both in magnitude and direction, and its vector coincides with the trajectory.
Unit of speed in system SI is determined by the length/time ratio, i.e. m/s .
Obviously, with curvilinear movement, the speed of the point will change in direction.
In order to establish the direction of the velocity vector at each moment of time during curvilinear motion, we divide the trajectory into infinitesimal sections of the path, which can be considered (due to their smallness) rectilinear. Then at each section the conditional speed v p
such a rectilinear motion will be directed along the chord, and the chord, in turn, with an infinite decrease in the length of the arc ( Δs
tends to zero) will coincide with the tangent to this arc.
It follows from this that during curvilinear motion the velocity vector at each moment of time coincides with the tangent to the trajectory (Fig. 1a). Rectilinear motion can be represented as a special case of curvilinear motion along an arc whose radius tends to infinity (trajectory coincides with tangent).
When a point moves unevenly, the magnitude of its velocity changes over time.
Let's imagine a point whose movement is given in a natural way by the equation s = f(t)
.
If in a short period of time Δt the point has passed the way Δs , then its average speed is:
vav = Δs/Δt.
Average speed does not give an idea of the true speed at any given moment in time (true speed is also called instantaneous speed). Obviously, the shorter the time period for which the average speed is determined, the closer its value will be to the instantaneous speed.
True (instantaneous) speed is the limit to which the average speed tends as Δt tends to zero:
v = lim v av at t→0 or v = lim (Δs/Δt) = ds/dt.
Thus, the numerical value of the true speed is v = ds/dt
.
The true (instantaneous) speed for any movement of a point is equal to the first derivative of the coordinate (i.e., the distance from the origin of the movement) with respect to time.
At Δt tending to zero, Δs also tends to zero, and, as we have already found out, the velocity vector will be directed tangentially (i.e., it coincides with the true velocity vector v ). It follows from this that the limit of the conditional speed vector v p , equal to the limit of the ratio of the point's displacement vector to an infinitesimal period of time, is equal to the vector of the point's true speed.
Fig.1
Let's look at an example. If a disk, without rotating, can slide along an axis fixed in a given reference system (Fig. 1, A), then in a given reference frame it obviously has only one degree of freedom - the position of the disk is uniquely determined, say, by the x coordinate of its center, measured along the axis. But if the disk, in addition, can also rotate (Fig. 1, b), then it acquires one more degree of freedom - to the coordinate x the rotation angle φ of the disk around the axis is added. If the axis with the disk is clamped in a frame that can rotate around a vertical axis (Fig. 1, V), then the number of degrees of freedom becomes equal to three - to x and φ the frame rotation angle is added ϕ .
A free material point in space has three degrees of freedom: for example, Cartesian coordinates x, y And z. The coordinates of a point can also be determined in cylindrical ( r, 𝜑, z) and spherical ( r, 𝜑, 𝜙) reference systems, but the number of parameters that uniquely determine the position of a point in space is always three.
A material point on a plane has two degrees of freedom. If we select a coordinate system in the plane xOy, then the coordinates x And y determine the position of a point on the plane, coordinate z is identically equal to zero.
A free material point on a surface of any kind has two degrees of freedom. For example: the position of a point on the Earth's surface is determined by two parameters: latitude and longitude.
A material point on a curve of any type has one degree of freedom. The parameter that determines the position of a point on a curve can be, for example, the distance along the curve from the origin.
Consider two material points in space connected by a rigid rod of length l(Fig. 2). The position of each point is determined by three parameters, but a connection is imposed on them.
Fig.2
The equation l 2 =(x 2 -x 1) 2 +(y 2 -y 1) 2 +(z 2 -z 1) 2 is the coupling equation. From this equation, any one coordinate can be expressed in terms of the other five coordinates (five independent parameters). Therefore, these two points have (2∙3-1=5) five degrees of freedom.
Let us consider three material points in space that do not lie on the same straight line, connected by three rigid rods. The number of degrees of freedom of these points is (3∙3-3=6) six.
A free rigid body generally has 6 degrees of freedom. Indeed, the position of a body in space relative to any reference system is determined by specifying three of its points that do not lie on the same straight line, and the distances between points in a rigid body remain unchanged during any of its movements. According to the above, the number of degrees of freedom should be six.
Forward movement
In kinematics, as in statistics, we will consider all rigid bodies as absolutely rigid.
Absolutely solid body is a material body whose geometric shape and dimensions do not change under any mechanical influences from other bodies, and the distance between any two of its points remains constant.
Kinematics of a rigid body, as well as the dynamics of a rigid body, is one of the most difficult sections of the course in theoretical mechanics.
Rigid body kinematics problems fall into two parts:
1) setting the movement and determining the kinematic characteristics of the movement of the body as a whole;
2) determination of the kinematic characteristics of the movement of individual points of the body.
There are five types of rigid body motion:
1) forward movement;
2) rotation around a fixed axis;
3) flat movement;
4) rotation around a fixed point;
5) free movement.
The first two are called the simplest motions of a rigid body.
Let's start by considering the translational motion of a rigid body.
Progressive is the movement of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.
Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines. Let's give examples.
1. The car body on a straight horizontal section of the road moves forward. In this case, the trajectories of its points will be straight lines.
2. Sparnik AB(Fig. 3) when the cranks O 1 A and O 2 B rotate, they also move translationally (any straight line drawn in it remains parallel to its initial direction). The points of the partner move in circles.
Fig.3
The pedals of a bicycle move progressively relative to its frame during movement, the pistons in the cylinders of an internal combustion engine move relative to the cylinders, and the cabins of Ferris wheels in parks (Fig. 4) relative to the Earth.
Fig.4
The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.
To prove this, consider a rigid body undergoing translational motion relative to the reference frame Oxyz. Let's take two arbitrary points in the body A And IN, whose positions at the moment of time t are determined by radius vectors and (Fig. 5).
Fig.5
Let's draw a vector connecting these points.
In this case, the length AB constant, like the distance between points of a rigid body, and the direction AB remains unchanged as the body moves forward. So the vector AB remains constant throughout the body's movement ( AB=const). As a result, the trajectory of point B is obtained from the trajectory of point A by parallel displacement of all its points by a constant vector. Therefore, the trajectories of the points A And IN will really be the same (when superimposed, coinciding) curves.
To find the velocities of points A And IN Let us differentiate both sides of the equality with respect to time. We get
But the derivative of a constant vector AB equal to zero. Derivatives of vectors and with respect to time give the velocities of points A And IN. As a result, we find that
those. what are the speeds of the points A And IN bodies at any moment of time are identical both in magnitude and direction. Taking derivatives with respect to time from both sides of the resulting equality:
Therefore, the accelerations of the points A And IN bodies at any moment of time are also identical in magnitude and direction.
Since the points A And IN were chosen arbitrarily, then from the results found it follows that for all points of the body their trajectories, as well as velocities and accelerations at any time, will be the same. Thus, the theorem is proven.
It follows from the theorem that the translational motion of a rigid body is determined by the movement of any one of its points. Consequently, the study of the translational motion of a body comes down to the problem of the kinematics of a point, which we have already considered.
During translational motion, the speed common to all points of the body is called the speed of translational motion of the body, and acceleration is called the acceleration of translational motion of the body. Vectors and can be depicted as applied at any point of the body.
Note that the concept of speed and acceleration of a body makes sense only in translational motion. In all other cases, the points of the body, as we will see, move with different speeds and accelerations, and the terms<<скорость тела>> or<<ускорение тела>> these movements lose their meaning.
Fig.6
During the time ∆t, the body, moving from point A to point B, makes a displacement equal to the chord AB, and covers a path equal to the length of the arc l.
The radius vector rotates through an angle ∆φ. The angle is expressed in radians.
The speed of movement of a body along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval ∆t during which this arc is passed:
A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called angular velocity:
The SI unit of angular velocity is radian per second.
With uniform motion in a circle, the angular velocity and the linear velocity module are constant values: ω=const; v=const.
The position of the body can be determined if the modulus of the radius vector and the angle φ it makes with the Ox axis (angular coordinate) are known. If at the initial moment of time t 0 =0 the angular coordinate is equal to φ 0, and at the moment of time t it is equal to φ, then the angle of rotation ∆φ of the radius vector during the time ∆t=t-t 0 is equal to ∆φ=φ-φ 0. Then from the last formula we can obtain the kinematic equation of motion of a material point in a circle:
It allows you to determine the position of the body at any time t.
Considering that , we get:
Formula for the relationship between linear and angular velocity.
The time period T during which the body makes one full revolution is called the period of rotation:
Where N is the number of revolutions made by the body during time Δt.
During the time ∆t=T the body travels the path l=2πR. Hence,
At ∆t→0, the angle is ∆φ→0 and, therefore, β→90°. The perpendicular to the tangent to the circle is the radius. Therefore, it is directed radially towards the center and is therefore called centripetal acceleration:
Module , direction changes continuously (Fig. 8). Therefore, this movement is not uniformly accelerated.
Fig.8
Fig.9
Then the position of the body at any moment of time is uniquely determined by the angle φ between these half-planes taken with the appropriate sign, which we will call the angle of rotation of the body. We will consider the angle φ to be positive if it is plotted from the fixed plane in a counterclockwise direction (for an observer looking from the positive end of the Az axis), and negative if it is clockwise. We will always measure the angle φ in radians. To know the position of the body at any moment in time, you need to know the dependence of the angle φ on time t, i.e.
The equation expresses the law of rotational motion of a rigid body around a fixed axis.
During rotational motion of an absolutely rigid body around a fixed axis the angles of rotation of the radius vector of different points of the body are the same.
The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity ω and angular acceleration ε.
If during a period of time ∆t=t 1 -t the body rotates through an angle ∆φ=φ 1 -φ, then the numerically average angular velocity of the body during this period of time will be . In the limit at ∆t→0 we find that
Thus, the numerical value of the angular velocity of a body at a given time is equal to the first derivative of the angle of rotation with respect to time. The sign of ω determines the direction of rotation of the body. It is easy to see that when rotation occurs counterclockwise, ω>0, and when clockwise, then ω<0.
The dimension of angular velocity is 1/T (i.e. 1/time); the unit of measurement is usually rad/s or, which is the same, 1/s (s -1), since the radian is a dimensionless quantity.
The angular velocity of a body can be represented as a vector whose modulus is equal to | | and which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise (Fig. 10). Such a vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.
Fig.10
The angle of rotation and angular velocity characterize the motion of the entire absolutely rigid body as a whole. The linear speed of any point on an absolutely rigid body is proportional to the distance of the point from the axis of rotation:
With uniform rotation of an absolutely rigid body, the angles of rotation of the body for any equal periods of time are the same, there are no tangential accelerations at various points of the body, and the normal acceleration of a point of the body depends on its distance to the axis of rotation:
The vector is directed along the radius of the point's trajectory towards the axis of rotation.
Angular acceleration characterizes the change in the angular velocity of a body over time. If over a period of time ∆t=t 1 -t the angular velocity of a body changes by the amount ∆ω=ω 1 -ω, then the numerical value of the average angular acceleration of the body over this period of time will be . In the limit at ∆t→0 we find,
Thus, the numerical value of the angular acceleration of a body at a given time is equal to the first derivative of the angular velocity or the second derivative of the angle of rotation of the body with respect to time.
The dimension of angular acceleration is 1/T 2 (1/time 2); the unit of measurement is usually rad/s 2 or, what is the same, 1/s 2 (s-2).
If the module of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, it is called slow. It is easy to see that the rotation will be accelerated when the quantities ω and ε have the same signs, and slowed down when they are different.
The angular acceleration of a body (by analogy with angular velocity) can also be represented as a vector ε directed along the axis of rotation. Wherein
The direction of ε coincides with the direction of ω when the body rotates at an accelerated rate (Fig. 10, a), and is opposite to ω when the body rotates at a slow speed (Fig. 10, b).
Fig.11 Fig. 12
2. Acceleration of body points. To find the acceleration of a point M let's use the formulas
In our case ρ=h. Substituting the value v into the expressions a τ and a n, we get:
or finally:
The tangential component of acceleration a τ is directed tangentially to the trajectory (in the direction of motion during accelerated rotation of the body and in the opposite direction during slow rotation); the normal component a n is always directed along the radius MS to the axis of rotation (Fig. 12). Total point acceleration M will
The deviation of the total acceleration vector from the radius of the circle described by the point is determined by the angle μ, which is calculated by the formula
Substituting the values of a τ and a n here, we get
Since ω and ε have the same value for all points of the body at a given moment in time, the accelerations of all points of a rotating rigid body are proportional to their distances from the axis of rotation and form at a given moment in time the same angle μ with the radii of the circles they describe . The acceleration field of points of a rotating rigid body has the form shown in Fig. 14.
Fig.13 Fig.14
3. Vectors of velocity and acceleration of body points. To find expressions directly for vectors v and a, let’s draw from an arbitrary point ABOUT axes AB radius vector of a point M(Fig. 13). Then h=r∙sinα and by the formula
So I can
The mechanical movement of a body is the change in its position in space relative to other bodies over time. He studies the movement of mechanic bodies. The motion of an absolutely rigid body (not deformed during movement and interaction), in which all its points at a given moment in time move equally, is called translational motion; to describe it, it is necessary and sufficient to describe the motion of one point of the body. A movement in which the trajectories of all points of the body are circles with a center on one line and all planes of the circles are perpendicular to this line is called rotational movement. A body whose shape and dimensions can be neglected under given conditions is called a material point. This is neglected
It is permissible to do this when the size of the body is small compared to the distance it travels or the distance of the body to other bodies. To describe the movement of a body, you need to know its coordinates at any moment in time. This is the main task of mechanics.
2. Relativity of motion. Reference system. Units.
To determine the coordinates of a material point, it is necessary to select a reference body and associate a coordinate system with it and set the origin of time. The coordinate system and the indication of the origin of time form a reference system relative to which the movement of the body is considered. The system must move at a constant speed (or be at rest, which is generally the same thing). The trajectory of the body, the distance traveled and displacement depend on the choice of the reference system, i.e. mechanical motion is relative. The unit of length is the meter, which is equal to the distance traveled by light in a vacuum in seconds. A second is a unit of time, equal to the periods of radiation of a cesium-133 atom.
3. Trajectory. Path and movement. Instant speed.
The trajectory of a body is a line described in space by a moving material point. Path – the length of the trajectory section from the initial to the final movement of the material point. Radius vector is a vector connecting the origin of coordinates and a point in space. Displacement is a vector connecting the starting and ending points of a trajectory section covered over time. Speed is a physical quantity that characterizes the speed and direction of movement at a given moment in time. The average speed is defined as. The average ground speed is equal to the ratio of the distance traveled by the body during a period of time to this interval. . Instantaneous speed (vector) is the first derivative of the radius vector of a moving point. . The instantaneous speed is directed tangentially to the trajectory, the average – along the secant. Instantaneous ground speed (scalar) – the first derivative of the track with respect to time, equal in magnitude to the instantaneous speed
4. Uniform linear movement. Graphs of kinematic quantities versus time in uniform motion. Addition of speeds.
Movement with a constant speed in magnitude and direction is called uniform rectilinear movement. With uniform rectilinear motion, a body travels equal distances in any equal periods of time. If the speed is constant, then the distance traveled is calculated as: The classical law of addition of velocities is formulated as follows: the speed of movement of a material point in relation to a reference system taken as a stationary one is equal to the vector sum of the speeds of movement of a point in a moving system and the speed of movement of a moving system relative to a stationary one. 
5. Acceleration. Uniformly accelerated linear motion. Graphs of the dependence of kinematic quantities on time in uniformly accelerated motion.
A movement in which a body makes unequal movements at equal intervals of time is called uneven movement. With uneven translational motion, the speed of the body changes over time. Acceleration (vector) is a physical quantity that characterizes the rate of change in speed in magnitude and direction. Instantaneous acceleration (vector) is the first derivative of speed with respect to time. .Uniformly accelerated is motion with acceleration that is constant in magnitude and direction. Velocity during uniformly accelerated motion is calculated as:
From here the formula for the path during uniformly accelerated motion is derived as
The formulas derived from the equations of speed and path for uniformly accelerated motion are also valid.
6. Free fall of bodies. Acceleration of gravity.
The fall of a body is its movement in the field of gravity (???) . The fall of bodies in a vacuum is called free fall. It has been experimentally established that during free fall, bodies move the same way regardless of their physical characteristics. The acceleration with which bodies fall to Earth in a vacuum is called the acceleration of free fall and is denoted
7. Uniform movement in a circle. Acceleration during uniform motion of a body in a circle (centripetal acceleration)
Any movement on a sufficiently small section of the trajectory can be approximately considered as a uniform movement in a circle. In the process of uniform motion around a circle, the speed value remains constant, but the direction of the speed vector changes.<рисунок>.. The acceleration vector when moving in a circle is directed perpendicular to the velocity vector (directed tangentially), to the center of the circle. The period of time during which a body makes a complete revolution around a circle is called a period. . The reciprocal of the period, showing the number of revolutions per unit time, is called frequency. Using these formulas, we can deduce that , or . Angular velocity (rotation speed) is defined as 
. The angular velocity of all points of the body is the same, and characterizes the movements of the rotating body as a whole. In this case, the linear velocity of the body is expressed as , and acceleration – as .
The principle of independence of movements considers the movement of any point of the body as the sum of two movements - translational and rotational. 
8. Newton's first law. Inertial reference system.
The phenomenon of maintaining the speed of a body in the absence of external influences is called inertia. Newton's first law, also known as the law of inertia, states: “there are such frames of reference relative to which translationally moving bodies maintain their speed constant unless other bodies act on them.” Reference systems relative to which bodies, in the absence of external influences, move rectilinearly and uniformly are called inertial reference systems. The reference systems associated with the earth are considered inertial, provided that the rotation of the earth is neglected.
9. Mass. Force. Newton's second law. Addition of forces. Center of gravity.
The reason for a change in the speed of a body is always its interaction with other bodies. When two bodies interact, the velocities always change, i.e. accelerations are acquired. The ratio of the accelerations of two bodies is the same for any interaction. The property of a body on which its acceleration when interacting with other bodies depends is called inertia. A quantitative measure of inertia is body weight. The ratio of the masses of interacting bodies is equal to the inverse ratio of the acceleration modules. Newton's second law establishes a connection between the kinematic characteristics of motion - acceleration, and the dynamic characteristics of interaction - forces. , or, in a more precise form, , i.e. the rate of change of momentum of a material point is equal to the force acting on it. When several forces are simultaneously applied to one body, the body moves with acceleration, which is the vector sum of the accelerations that would arise under the influence of each of these forces separately. The forces acting on a body and applied to one point are added according to the rule of vector addition. This position is called the principle of independence of forces. The center of mass is a point of a rigid body or system of rigid bodies that moves in the same way as a material point with a mass equal to the sum of the masses of the entire system as a whole, which is subject to the same resultant force as the body. 
. By integrating this expression over time, we can obtain expressions for the coordinates of the center of mass. The center of gravity is the point of application of the resultant of all gravity forces acting on the particles of this body at any position in space. If the linear dimensions of the body are small compared to the size of the Earth, then the center of mass coincides with the center of gravity. The sum of the moments of all forces of elementary gravity relative to any axis passing through the center of gravity is equal to zero.
10. Newton's third law.
For any interaction of two bodies, the ratio of the modules of the acquired accelerations is constant and equal to the inverse ratio of the masses. Because When bodies interact, the acceleration vectors have the opposite direction, we can write that 
. According to Newton's second law, the force acting on the first body is equal to , and on the second. Thus, . Newton's third law relates the forces with which bodies act on each other. If two bodies interact with each other, then the forces arising between them are applied to different bodies, are equal in magnitude, opposite in direction, act along the same straight line, and have the same nature.
11. Elastic forces. Hooke's law.
The force arising as a result of deformation of a body and directed in the direction opposite to the movement of particles of the body during this deformation is called elastic force. Experiments with a rod have shown that for small deformations compared to the size of the body, the modulus of the elastic force is directly proportional to the modulus of the displacement vector of the free end of the rod, which in projection looks like . This connection was established by R. Hooke; his law is formulated as follows: the elastic force that arises during deformation of a body is proportional to the elongation of the body in the direction opposite to the direction of movement of the particles of the body during deformation. Coefficient k called the rigidity of the body, and depends on the shape and material of the body. Expressed in newtons per meter. Elastic forces are caused by electromagnetic interactions.
12. Friction forces, sliding friction coefficient. Viscous friction (???)
The force that arises at the boundary of interaction of bodies in the absence of relative motion of the bodies is called the static friction force. The static friction force is equal in magnitude to the external force directed tangentially to the surface of contact of the bodies and opposite in direction. When one body moves uniformly over the surface of another under the influence of an external force, a force acts on the body that is equal in magnitude to the driving force and opposite in direction. This force is called sliding friction force. The sliding friction force vector is directed opposite the velocity vector, so this force always leads to a decrease in the relative speed of the body. Friction forces, like the elastic force, are of an electromagnetic nature, and arise due to the interaction between the electric charges of the atoms of contacting bodies. It has been experimentally established that the maximum value of the modulus of the static friction force is proportional to the pressure force. The maximum value of the static friction force and the sliding friction force are also approximately equal, as are the coefficients of proportionality between the friction forces and the pressure of the body on the surface.
13. Gravitational forces. The law of universal gravitation. Gravity. Body weight.
From the fact that bodies, regardless of their mass, fall with the same acceleration, it follows that the force acting on them is proportional to the mass of the body. This attractive force acting on all bodies from the Earth is called gravity. The force of gravity acts at any distance between bodies. All bodies attract each other, the force of universal gravity is directly proportional to the product of masses and inversely proportional to the square of the distance between them. The vectors of universal gravitational forces are directed along a straight line connecting the centers of mass of bodies. , G – Gravitational constant, equal to . Body weight is the force with which the body, due to gravity, acts on a support or stretches a suspension. The weight of the body is equal in magnitude and opposite in direction to the elastic force of the support according to Newton's third law. According to Newton's second law, if no force acts on a body anymore, then the force of gravity of the body is balanced by the force of elasticity. As a result, the weight of the body on a stationary or uniformly moving horizontal support is equal to the force of gravity. If the support moves with acceleration, then according to Newton’s second law 
, from where it is derived. This means that the weight of a body whose acceleration direction coincides with the direction of acceleration due to gravity is less than the weight of a body at rest.
14. Vertical movement of a body under the influence of gravity. Movement of artificial satellites. Weightlessness. First escape velocity.
When throwing a body parallel to the earth's surface, the greater the initial speed, the greater the flight range. At high speeds, it is also necessary to take into account the sphericity of the earth, which is reflected in a change in the direction of the gravity vector. At a certain speed, a body can move around the Earth under the influence of universal gravity. This speed, called the first cosmic speed, can be determined from the equation of motion of a body in a circle. On the other hand, from Newton's second law and the law of universal gravitation it follows that. So at a distance R from the center of a celestial body with mass M the first escape velocity is equal to. When the speed of a body changes, the shape of its orbit changes from a circle to an ellipse. When the second escape velocity is reached, the orbit becomes parabolic.
15. Body impulse. Law of conservation of momentum. Jet propulsion.
According to Newton's second law, regardless of whether a body was at rest or moving, a change in its speed can only occur when interacting with other bodies. If the body weighs m for a time t a force acts and the speed of its movement changes from to , then the acceleration of the body is equal to . Based on Newton's second law for force, we can write . A physical quantity equal to the product of a force and the time of its action is called the impulse of a force. The impulse of a force shows that there is a quantity that changes equally in all bodies under the influence of the same forces, if the time of action of the force is the same. This quantity, equal to the product of the mass of the body and the speed of its movement, is called the momentum of the body. The change in the momentum of the body is equal to the impulse of the force that caused this change. Let's take two bodies, with masses and , moving with velocities and . According to Newton's third law, the forces acting on bodies during their interaction are equal in magnitude and opposite in direction, i.e. they can be designated as and . For changes in impulses during interaction, we can write . From these expressions we get that 
, that is, the vector sum of the momenta of two bodies before the interaction is equal to the vector sum of the momenta after the interaction. In a more general form, the law of conservation of momentum sounds like this: If, then.
16. Mechanical work. Power. Kinetic and potential energy.
Work A a force constant is a physical quantity equal to the product of the force and displacement moduli multiplied by the cosine of the angle between the vectors and. 
. Work is a scalar quantity and can be negative if the angle between the displacement and force vectors is greater than . The unit of work is called the joule, 1 joule is equal to the work done by a force of 1 newton when moving the point of its application by 1 meter. Power is a physical quantity equal to the ratio of work to the period of time during which this work was performed. . A unit of power is called a watt; 1 watt is equal to the power at which 1 joule of work is done in 1 second. Let us assume that a body of mass m a force acts (which can generally be the resultant of several forces), under the influence of which the body moves in the direction of the vector . The modulus of force according to Newton's second law is equal to ma, and the magnitude of the displacement vector is related to the acceleration and the initial and final velocities. This gives us the formula to work with: 
. A physical quantity equal to half the product of body mass and the square of speed is called kinetic energy. The work done by the resultant forces applied to the body is equal to the change in kinetic energy. A physical quantity equal to the product of the mass of a body by the acceleration modulus of free fall and the height to which the body is raised above a surface with zero potential is called the potential energy of the body. The change in potential energy characterizes the work of gravity to move a body. This work is equal to the change in potential energy taken with the opposite sign. A body located below the surface of the earth has negative potential energy. Not only raised bodies have potential energy. Let us consider the work done by the elastic force when the spring is deformed. The elastic force is directly proportional to the deformation, and its average value will be equal to 
, work is equal to the product of force and deformation 
, or 
. A physical quantity equal to half the product of the rigidity of a body by the square of the deformation is called the potential energy of a deformed body. An important characteristic of potential energy is that a body cannot possess it without interacting with other bodies.
17. Laws of conservation of energy in mechanics.
Potential energy characterizes interacting bodies, kinetic energy characterizes moving bodies. Both arise as a result of the interaction of bodies. If several bodies interact with each other only by gravitational and elastic forces, and no external forces act on them (or their resultant is zero), then for any interactions of bodies, the work of the elastic or gravitational forces is equal to the change in potential energy taken with the opposite sign . At the same time, according to the kinetic energy theorem (the change in the kinetic energy of a body is equal to the work of external forces), the work of the same forces is equal to the change in kinetic energy. . From this equality it follows that the sum of the kinetic and potential energies of the bodies that make up a closed system and interact with each other by the forces of gravity and elasticity remains constant. The sum of the kinetic and potential energies of bodies is called total mechanical energy. The total mechanical energy of a closed system of bodies interacting with each other by the forces of gravity and elasticity remains unchanged. The work of the forces of gravity and elasticity is equal, on the one hand, to an increase in kinetic energy, and on the other, to a decrease in potential energy, that is, the work is equal to the energy converted from one type to another.
18. Simple mechanisms (inclined plane, lever, block) and their application.
An inclined plane is used so that a body of large mass can be moved by a force significantly less than the weight of the body. If the angle of the inclined plane is a, then to move the body along the plane it is necessary to apply a force equal to . The ratio of this force to the weight of the body, neglecting the friction force, is equal to the sine of the angle of inclination of the plane. But with a gain in strength, there is no gain in work, because the path increases several times. This result is a consequence of the law of conservation of energy, since the work done by gravity does not depend on the lifting trajectory of the body.
A lever is in equilibrium if the moment of forces rotating it clockwise is equal to the moment of forces rotating the lever counterclockwise. If the directions of the force vectors applied to the lever are perpendicular to the shortest straight lines connecting the points of application of the forces and the axis of rotation, then the equilibrium conditions take the form. If , then the lever provides a gain in strength. A gain in strength does not give a gain in work, because when turning through an angle a, the force does work, and the force does work. Because according to condition , then .
The block allows you to change the direction of the force. The shoulders of the forces applied to different points of the fixed block are the same, and therefore the fixed block does not provide any gain in strength. When lifting a load using a moving block, the gain in strength is doubled, because The gravity arm is half as large as the cable tension arm. But when pulling the cable to a length l the load rises to a height l/2 Therefore, a stationary block also does not provide any gain in work.
19. Pressure. Pascal's law for liquids and gases.
A physical quantity equal to the ratio of the modulus of the force acting perpendicular to the surface to the area of this surface is called pressure. The unit of pressure is the pascal, which is equal to the pressure produced by a force of 1 newton per area of 1 square meter. All liquids and gases transmit the pressure exerted on them in all directions.
20. Communicating vessels. Hydraulic Press. Atmosphere pressure. Bernoulli's equation.
In a cylindrical vessel, the pressure force on the bottom of the vessel is equal to the weight of the liquid column. The pressure at the bottom of the vessel is equal to 
, where does the pressure at depth come from? h equals . The same pressure acts on the walls of the vessel. The equality of liquid pressures at the same height leads to the fact that in communicating vessels of any shape, the free surfaces of a homogeneous liquid at rest are at the same level (in the case of negligible capillary forces). In the case of a non-uniform liquid, the height of the column of a denser liquid will be less than the height of a less dense liquid. A hydraulic machine operates based on Pascal's law. It consists of two communicating vessels, closed by pistons of different areas. The pressure produced by an external force on one piston is transferred according to Pascal's law to the second piston. 
. A hydraulic machine provides a gain in force as many times as the area of its large piston is greater than the area of the small one.
For stationary motion of an incompressible fluid, the continuity equation is valid. For an ideal fluid in which viscosity (i.e., friction between its particles) can be neglected, the mathematical expression for the law of conservation of energy is the Bernoulli equation 
.
21. Torricelli's experience. Change in atmospheric pressure with altitude.
Under the influence of gravity, the upper layers of the atmosphere press on the underlying ones. This pressure, according to Pascal's law, is transmitted in all directions. This pressure is greatest at the Earth's surface, and is determined by the weight of the air column from the surface to the boundary of the atmosphere. As altitude increases, the mass of atmospheric layers pressing on the surface decreases, therefore, atmospheric pressure decreases with altitude. At sea level, the atmospheric pressure is 101 kPa. This pressure is exerted by a column of mercury 760 mm high. If a tube in which a vacuum is created is lowered into liquid mercury, then under the influence of atmospheric pressure the mercury will rise in it to such a height at which the pressure of the liquid column becomes equal to the external atmospheric pressure on the open surface of the mercury. When atmospheric pressure changes, the height of the liquid column in the tube will also change.
22. Archimedes' power of the day of liquids and gases. Sailing conditions tel.
The dependence of pressure in liquids and gases on depth leads to the emergence of a buoyant force acting on any body immersed in a liquid or gas. This force is called Archimedean force. If a body is immersed in a liquid, then the pressures on the side walls of the vessel are balanced by each other, and the resultant of the pressures from below and above is the Archimedean force. , i.e. The force pushing out a body immersed in a liquid (gas) is equal to the weight of the liquid (gas) displaced by the body. The Archimedean force is directed opposite to the force of gravity, therefore, when weighed in a liquid, the weight of a body is less than in a vacuum. A body in a liquid is acted upon by gravity and the Archimedean force. If the force of gravity is greater in modulus, the body sinks; if it is less, it floats; if they are equal, it can be in equilibrium at any depth. These force ratios are equal to the ratio of the densities of the body and liquid (gas).
23. Basic principles of molecular kinetic theory and their experimental substantiation. Brownian motion. Weight and size molecules.
Molecular kinetic theory is the study of the structure and properties of matter, using the idea of the existence of atoms and molecules as the smallest particles of matter. The main provisions of MCT: matter consists of atoms and molecules, these particles move chaotically, the particles interact with each other. The movement of atoms and molecules and their interaction obeys the laws of mechanics. In the interaction of molecules when they approach each other, the forces of attraction first prevail. At a certain distance between them, repulsive forces arise that exceed the attractive forces in magnitude. Molecules and atoms oscillate randomly about positions where the forces of attraction and repulsion balance each other. In a liquid, molecules not only vibrate, but also jump from one equilibrium position to another (fluidity). In gases, the distances between atoms are much larger than the sizes of molecules (compressibility and expansion). R. Brown discovered at the beginning of the 19th century that solid particles move randomly in a liquid. This phenomenon could only be explained by MCT. Randomly moving molecules of a liquid or gas collide with a solid particle and change the direction and speed of its movement (while, of course, changing both its direction and speed). The smaller the particle size, the more noticeable the change in momentum becomes. Any substance consists of particles, therefore the amount of substance is considered to be proportional to the number of particles. The unit of quantity of a substance is called the mole. A mole is equal to the amount of a substance containing as many atoms as there are in 0.012 kg of carbon 12 C. The ratio of the number of molecules to the amount of substance is called Avogadro’s constant: 
. The amount of a substance can be found as the ratio of the number of molecules to Avogadro's constant. Molar mass M is a quantity equal to the ratio of the mass of a substance m to the amount of substance. Molar mass is expressed in kilograms per mole. Molar mass can be expressed in terms of the mass of the molecule m 0 : 
.
24. Ideal gas. Basic equation of the molecular kinetic theory of an ideal gas.
To explain the properties of matter in the gaseous state, the ideal gas model is used. This model assumes the following: gas molecules are negligibly small compared to the volume of the vessel, there are no attractive forces between the molecules, and when they collide with each other and the walls of the vessel, repulsive forces act. A qualitative explanation of the phenomenon of gas pressure is that molecules of an ideal gas, when colliding with the walls of a vessel, interact with them as elastic bodies. When a molecule collides with the wall of a vessel, the projection of the velocity vector onto the axis perpendicular to the wall changes to the opposite. Therefore, during a collision, the velocity projection varies from –mv x before mv x, and the change in momentum is . During a collision, the molecule acts on the wall with a force equal, according to Newton's third law, to the force opposite in direction. There are a lot of molecules, and the average value of the geometric sum of the forces acting on the part of individual molecules forms the force of gas pressure on the walls of the vessel. Gas pressure is equal to the ratio of the modulus of pressure force to the area of the vessel wall: p=F/S. Let us assume that the gas is in a cubic container. The momentum of one molecule is 2 mv, one molecule acts on the wall with an average force 2mv/Dt. Time D t the movement from one wall of the vessel to the other is equal to 2l/v, hence, . The force of pressure on the wall of the vessel of all molecules is proportional to their number, i.e. 
. Due to the complete randomness of the movement of molecules, their movement in each direction is equally probable and equal to 1/3 of the total number of molecules. Thus, . Since the pressure is applied to the face of a cube with an area l 2, then the pressure will be equal. This equation is called the basic equation of molecular kinetic theory. Denoting the average kinetic energy of molecules, we obtain.
25. Temperature, its measurement. Absolute temperature scale. Speed of gas molecules.
The basic MKT equation for an ideal gas establishes a connection between micro- and macroscopic parameters. When two bodies come into contact, their macroscopic parameters change. When this change has ceased, thermal equilibrium is said to have occurred. A physical parameter that is the same in all parts of a system of bodies in a state of thermal equilibrium is called body temperature. Experiments have shown that for any gas in a state of thermal equilibrium, the ratio of the product of pressure and volume to the number of molecules is the same 
. This allows the value to be taken as a measure of temperature. Because n=N/V, then taking into account the basic MKT equation, therefore, the value is equal to two-thirds of the average kinetic energy of the molecules. , Where k– proportionality coefficient depending on the scale. On the left side of this equation the parameters are non-negative. Hence, the temperature of a gas at which its pressure at a constant volume is zero is called absolute zero temperature. The value of this coefficient can be found from two known states of matter with known pressure, volume, number of molecules and temperature. . Coefficient k, called Boltzmann's constant, is equal to 
. From the equations for the relationship between temperature and average kinetic energy it follows, i.e. the average kinetic energy of the chaotic movement of molecules is proportional to the absolute temperature. , . This equation shows that at the same temperature and concentration of molecules, the pressure of any gases is the same.
26. Equation of state of an ideal gas (Mendeleev-Clapeyron equation). Isothermal, isochoric and isobaric processes.
Using the dependence of pressure on concentration and temperature, one can find the relationship between the macroscopic parameters of a gas - volume, pressure and temperature. . This equation is called the ideal gas equation of state (Mendeleev-Clapeyron equation).
An isothermal process is a process that occurs at a constant temperature. From the equation of state of an ideal gas it follows that at constant temperature, mass and composition of the gas, the product of pressure and volume must remain constant. The graph of an isotherm (curve of an isothermal process) is a hyperbola. The equation is called the Boyle-Mariotte law.
An isochoric process is a process that occurs at a constant volume, mass and composition of the gas. Under these conditions 
, where is the temperature coefficient of gas pressure. This equation is called Charles's law. The graph of the equation of an isochoric process is called an isochore, and is a straight line passing through the origin.
An isobaric process is a process that occurs at constant pressure, mass and composition of the gas. In the same way as for an isochoric process, we can obtain an equation for an isobaric process 
. The equation that describes this process is called Gay-Lussac's law. The graph of the equation of an isobaric process is called an isobar, and is a straight line passing through the origin of coordinates.
27. Internal energy. Work in thermodynamics.
If the potential energy of interaction of molecules is zero, then the internal energy is equal to the sum of the kinetic energies of motion of all gas molecules 
. Consequently, when the temperature changes, the internal energy of the gas also changes. Substituting the equation of state of an ideal gas into the energy equation, we find that the internal energy is directly proportional to the product of gas pressure and volume. . The internal energy of a body can only change when interacting with other bodies. During mechanical interaction of bodies (macroscopic interaction), the measure of transferred energy is work A. During heat exchange (microscopic interaction), the measure of energy transferred is the amount of heat Q. In a non-isolated thermodynamic system, the change in internal energy D U equal to the sum of the transferred amount of heat Q and the work of external forces A. Instead of work A performed by external forces, it is more convenient to consider the work A` performed by the system over external bodies. A=–A`. Then the first law of thermodynamics is expressed as, or. This means that any machine can perform work on external bodies only by receiving an amount of heat from outside Q or decrease in internal energy D U. This law excludes the creation of a perpetual motion machine of the first kind.
28. Amount of heat. Specific heat capacity of a substance. The law of conservation of energy in thermal processes (the first law of thermodynamics).
The process of transferring heat from one body to another without doing work is called heat transfer. The energy transferred to the body as a result of heat exchange is called the amount of heat. If the heat transfer process is not accompanied by work, then it is based on the first law of thermodynamics. The internal energy of a body is proportional to the mass of the body and its temperature, therefore 
. Magnitude With is called specific heat capacity, the unit is . Specific heat capacity shows how much heat must be transferred to heat 1 kg of a substance by 1 degree. Specific heat capacity is not an unambiguous characteristic and depends on the work done by the body during heat transfer.
When carrying out heat exchange between two bodies under conditions of zero work of external forces and in thermal isolation from other bodies, according to the law of conservation of energy 
. If the change in internal energy is not accompanied by work, then , or , where . This equation is called the heat balance equation.
29. Application of the first law of thermodynamics to isoprocesses. Adiabatic process. Irreversibility of thermal processes.
One of the main processes that perform work in most machines is the process of expansion of gas with the performance of work. If during isobaric expansion of a gas from volume V 1 up to volume V 2 the displacement of the cylinder piston was l, then work A perfect by the gas is equal to , or 
. If we compare the areas under the isobar and isotherm, which are work, we can conclude that with the same expansion of the gas at the same initial pressure in the case of an isothermal process, less work will be done. In addition to isobaric, isochoric and isothermal processes, there is the so-called. adiabatic process. Adiabatic is a process that occurs in the absence of heat transfer. The process of rapid expansion or compression of a gas can be considered close to adiabatic. In this process, work is done due to changes in internal energy, i.e. , therefore, during an adiabatic process the temperature decreases. Since during adiabatic compression of a gas the temperature of the gas increases, the pressure of the gas increases faster with a decrease in volume than during an isothermal process.
Heat transfer processes spontaneously occur in only one direction. Heat transfer always occurs to a colder body. The second law of thermodynamics states that a thermodynamic process is impossible, as a result of which heat would be transferred from one body to another, hotter one, without any other changes. This law excludes the creation of a perpetual motion machine of the second kind.
30. The principle of operation of heat engines. Heat engine efficiency.
Typically in heat engines, work is done by an expanding gas. The gas that does work during expansion is called the working fluid. Gas expansion occurs as a result of an increase in its temperature and pressure when heated. A device from which the working fluid receives heat Q called a heater. The device to which the machine transfers heat after completing its working stroke is called a refrigerator. First, the pressure increases isochorically, expands isobarically, cools isochorically, and contracts isobarically.<рисунок с подъемником>. As a result of the working cycle, the gas returns to its initial state, its internal energy takes on its original value. It means that . According to the first law of thermodynamics, . The work done by the body per cycle is equal to Q. The amount of heat received by the body per cycle is equal to the difference between that received from the heater and given to the refrigerator. Hence, . The efficiency of a machine is the ratio of useful energy used to expended energy. 
.
31. Evaporation and condensation. Saturated and unsaturated pairs. Air humidity.
The uneven distribution of kinetic energy of thermal motion leads to this. That at any temperature the kinetic energy of some of the molecules can exceed the potential binding energy with the rest. Evaporation is the process by which molecules escape from the surface of a liquid or solid. Evaporation is accompanied by cooling, because faster molecules leave the liquid. The evaporation of a liquid in a closed vessel at a constant temperature leads to an increase in the concentration of molecules in the gaseous state. After some time, an equilibrium occurs between the number of molecules evaporating and those returning to the liquid. A gaseous substance in dynamic equilibrium with its liquid is called saturated vapor. Steam at a pressure below the saturated vapor pressure is called unsaturated. Saturated vapor pressure does not depend on volume at constant temperature (from ). At a constant concentration of molecules, the pressure of saturated vapor increases faster than the pressure of an ideal gas, because Under the influence of temperature, the number of molecules increases. The ratio of water vapor pressure at a given temperature to saturated vapor pressure at the same temperature, expressed as a percentage, is called relative humidity. The lower the temperature, the lower the saturated vapor pressure, so when cooled to a certain temperature, the vapor becomes saturated. This temperature is called the dew point tp.
32. Crystalline and amorphous bodies. Mechanical properties of solids. Elastic deformations.
Amorphous bodies are those whose physical properties are the same in all directions (isotropic bodies). The isotropy of physical properties is explained by the random arrangement of molecules. Solids in which the molecules are ordered are called crystals. The physical properties of crystalline bodies are not the same in different directions (anisotropic bodies). The anisotropy of the properties of crystals is explained by the fact that with an ordered structure, the interaction forces are unequal in different directions. An external mechanical effect on a body causes a displacement of atoms from an equilibrium position, which leads to a change in the shape and volume of the body - deformation. Deformation can be characterized by absolute elongation, equal to the difference in lengths before and after deformation, or by relative elongation. When a body deforms, elastic forces arise. A physical quantity equal to the ratio of the modulus of elastic force to the cross-sectional area of a body is called mechanical stress. At small deformations, stress is directly proportional to elongation. Proportionality factor E in the equation is called the modulus of elasticity (Young's modulus). The modulus of elasticity is constant for a given material 
, where . The potential energy of a deformed body is equal to the work expended in tension or compression. From here 
.
Hooke's law holds true only for small deformations. The maximum voltage at which it is still satisfied is called the proportional limit. Beyond this limit, the voltage stops growing proportionally. Up to a certain level of stress, the deformed body will restore its dimensions after the load is removed. This point is called the elastic limit of the body. When the elastic limit is exceeded, plastic deformation begins, in which the body does not restore its previous shape. In the region of plastic deformation, the stress almost does not increase. This phenomenon is called material flow. Beyond the yield point, the stress increases to a point called the ultimate strength, after which the stress decreases until the body fails.
33. Properties of liquids. Surface tension. Capillary phenomena.
The possibility of free movement of molecules in a liquid determines the fluidity of the liquid. A body in a liquid state does not have a constant shape. The shape of the liquid is determined by the shape of the vessel and surface tension forces. Inside the liquid, the attractive forces of molecules are compensated, but at the surface they are not. Any molecule located near the surface is attracted by molecules inside the liquid. Under the influence of these forces, the molecules on the surface are pulled inward until the free surface becomes the smallest possible. Because If a sphere has the minimum surface for a given volume, then with little action of other forces the surface takes the form of a spherical segment. The surface of the liquid at the edge of the vessel is called the meniscus. The wetting phenomenon is characterized by the contact angle between the surface and the meniscus at the intersection point. The magnitude of the surface tension force on a section of length D l equal to . The curvature of the surface creates excess pressure on the liquid, equal to, for a known contact angle and radius 
. The coefficient s is called the surface tension coefficient. A capillary is a tube with a small internal diameter. With complete wetting, the surface tension force is directed along the surface of the body. In this case, the rise of the liquid through the capillary continues under the influence of this force until the force of gravity balances the force of surface tension, because , That .
34. Electric charge. Interaction of charged bodies. Coulomb's law. Law of conservation of electric charge.
Neither mechanics nor MCT is able to explain the nature of the forces that bind atoms. The laws of interaction of atoms and molecules can be explained on the basis of the concept of electric charges.<Опыт с натиранием ручки и притяжением бумажки>The interaction of bodies detected in this experiment is called electromagnetic, and is determined by electric charges. The ability of charges to attract and repel is explained by the assumption that there are two types of charges - positive and negative. Bodies charged with the same charge repel, but bodies with different charges attract. The unit of charge is a coulomb - a charge passing through the cross-section of a conductor in 1 second at a current of 1 ampere. In a closed system, into which electric charges do not enter from the outside and from which electric charges do not leave during any interactions, the algebraic sum of the charges of all bodies is constant. The basic law of electrostatics, also known as Coulomb's law, states that the modulus of the interaction force between two charges is directly proportional to the product of the moduli of the charges and inversely proportional to the square of the distance between them. The force is directed along the straight line connecting the charged bodies. It is a repulsive or attractive force, depending on the sign of the charges. Constant k in the expression of Coulomb's law is equal to 
. Instead of this coefficient, the so-called electrical constant associated with the coefficient k expression , from . The interaction of stationary electric charges is called electrostatic.
35. Electric field. Electric field strength. The principle of superposition of electric fields.
Based on the theory of short-range action, there is an electric field around each charge. An electric field is a material object, constantly exists in space and is capable of acting on other charges. An electric field propagates through space at the speed of light. A physical quantity equal to the ratio of the force with which the electric field acts on a test charge (a point positive small charge that does not affect the field configuration) to the value of this charge is called the electric field strength. Using Coulomb's law it is possible to obtain a formula for the field strength created by the charge q on distance r from charge 
. The field strength does not depend on the charge on which it acts. If on charge q Electric fields of several charges act simultaneously, then the resulting force turns out to be equal to the geometric sum of the forces acting from each field separately. This is called the principle of superposition of electric fields. The electric field intensity line is a line whose tangent at each point coincides with the intensity vector. Tension lines begin on positive charges and end on negative charges, or go to infinity. An electric field whose strength is the same for everyone at any point in space is called a uniform electric field. The field between two parallel oppositely charged metal plates can be considered approximately uniform. With uniform charge distribution q over the surface of the area S the surface charge density is . For an infinite plane with surface charge density s, the field strength is the same at all points in space and is equal to 
.
36. The work of the electrostatic field when moving a charge. Potential difference.
When a charge is moved by an electric field over a distance, the work done is equal to 
. As in the case of the work of gravity, the work of the Coulomb force does not depend on the trajectory of the charge. When the direction of the displacement vector changes by 180 0, the work of the field forces changes sign to the opposite. Thus, the work done by the electrostatic field forces when moving a charge along a closed circuit is zero. A field whose work of forces along a closed path is zero is called a potential field.
Just like a body of mass m in a gravity field has potential energy proportional to the mass of the body, an electric charge in an electrostatic field has potential energy W p, proportional to the charge. The work done by the electrostatic field forces is equal to the change in the potential energy of the charge, taken with the opposite sign. At one point in an electrostatic field, different charges can have different potential energies. But the ratio of potential energy to charge for a given point is a constant value. This physical quantity is called electric field potential, from which the potential energy of a charge is equal to the product of the potential at a given point and the charge. Potential is a scalar quantity; the potential of several fields is equal to the sum of the potentials of these fields. The measure of the change in energy during the interaction of bodies is work. When moving a charge, the work done by the electrostatic field forces is equal to the change in energy with the opposite sign, therefore. Because work depends on the potential difference and does not depend on the trajectory between them, then the potential difference can be considered an energy characteristic of the electrostatic field. If the potential at an infinite distance from the charge is taken equal to zero, then at a distance r from the charge it is determined by the formula .
The ratio of the work done by any electric field when moving a positive charge from one point of the field to another to the value of the charge is called the voltage between these points, where the work comes from. In an electrostatic field, the voltage between any two points is equal to the potential difference between these points. The unit of voltage (and potential difference) is called the volt. 1 volt is equal to the voltage at which the field does 1 joule of work to move 1 coulomb of charge. On the one hand, the work done to move a charge is equal to the product of force and displacement. On the other hand, it can be found from the known voltage between sections of the path. From here. The unit of electric field strength is volt per meter ( i/m).
A capacitor is a system of two conductors separated by a dielectric layer, the thickness of which is small compared to the size of the conductors. Between the plates the field strength is equal to twice the strength of each of the plates; outside the plates it is zero. A physical quantity equal to the ratio of the charge of one of the plates to the voltage between the plates is called the electrical capacity of the capacitor. The unit of electrical capacity is the farad; a capacitor has a capacity of 1 farad, between the plates of which the voltage is equal to 1 volt when a charge of 1 coulomb is imparted to the plates. The field strength between the plates of a solid capacitor is equal to the sum of the strength of the plates. 
, and because for a homogeneous field is satisfied, then 
, i.e. electrical capacity is directly proportional to the area of the plates and inversely proportional to the distance between them. When a dielectric is introduced between the plates, its electrical capacity increases by e times, where e is the dielectric constant of the introduced material.
38. The dielectric constant. Electric field energy.
Dielectric constant is a physical quantity that characterizes the ratio of the modulus of the electric field strength in a vacuum to the modulus of the electric field in a homogeneous dielectric. The work done by the electric field is equal, but when the capacitor is charged, its voltage increases from 0
before U, That's why 
. Therefore, the potential energy of the capacitor is equal to .
39. Electric current. Current strength. Conditions for the existence of electric current.
Electric current is the orderly movement of electric charges. The direction of the current is taken to be the movement of positive charges. Electric charges can move in an orderly manner under the influence of an electric field. Therefore, a sufficient condition for the existence of a current is the presence of a field and free charge carriers. An electric field can be created by two differently charged bodies connected. Charge ratio D q, transferred through the cross section of the conductor during the time interval D t to this interval is called the current strength. If the current strength does not change over time, then the current is called constant. In order for current to exist in a conductor for a long time, it is necessary that the conditions causing the current remain unchanged.<схема с один резистором и батареей>. The forces that cause charge to move inside a current source are called extraneous forces. In a galvanic cell (and any battery – g.e.???) they are the forces of a chemical reaction, in a DC machine - the Lorentz force.
40. Ohm's law for a section of a circuit. Conductor resistance. Dependence of conductor resistance on temperature. Superconductivity. Serial and parallel connection of conductors.
The ratio of the voltage between the ends of a section of an electrical circuit to the current is a constant value and is called resistance. The unit of resistance is 0 ohm; a resistance of 1 ohm is that section of the circuit in which, at a current of 1 ampere, the voltage is equal to 1 volt. Resistance is directly proportional to length and inversely proportional to cross-sectional area, where r is the electrical resistivity, a constant value for a given substance under given conditions. When heated, the resistivity of metals increases according to a linear law, where r 0 is the resistivity at 0 0 C, a is the temperature coefficient of resistance, specific for each metal. At temperatures close to absolute zero, the resistance of substances drops sharply to zero. This phenomenon is called superconductivity. The passage of current in superconducting materials occurs without loss of heating of the conductor.
Ohm's law for a section of a circuit is called the equation. When conductors are connected in series, the current is the same in all conductors, and the voltage at the ends of the circuit is equal to the sum of the voltages on all conductors connected in series. . When conductors are connected in series, the total resistance is equal to the sum of the resistances of the components. In a parallel connection, the voltage at the ends of each section of the circuit is the same, and the current strength is branched into separate parts. From here. When connecting conductors in parallel, the reciprocal value of the total resistance is equal to the sum of the reciprocal values of the resistances of all parallel-connected conductors.
41. Work and current power. Electromotive force. Ohm's law for a complete circuit.
The work done by the forces of the electric field that creates an electric current is called the work of the current. Job A current in the area with resistance R in time D t equal to . The power of the electric current is equal to the ratio of work to the time of completion, i.e. 
. Work is expressed, as usual, in joules, power - in watts. If no work is done on a section of the circuit under the influence of an electric field and no chemical reactions occur, then the work leads to heating of the conductor. In this case, the work is equal to the amount of heat released by the current-carrying conductor (Joule-Lenz Law).
In an electrical circuit, work is performed not only in the external section, but also in the battery. The electrical resistance of a current source is called internal resistance r. In the internal section of the circuit, an amount of heat equal to . The total work done by the forces of the electrostatic field when moving along a closed loop is zero, so all the work is done due to external forces that maintain a constant voltage. The ratio of the work done by external forces to the transferred charge is called electromotive force of the source, where D q– transferred charge. If, as a result of the passage of direct current, only heating of the conductors occurred, then according to the law of conservation of energy 
, i.e. . The current flow in an electrical circuit is directly proportional to the emf and inversely proportional to the total resistance of the circuit.
42. Semiconductors. Electrical conductivity of semiconductors and its dependence on temperature. Intrinsic and impurity conductivity of semiconductors.
Many substances do not conduct current as well as metals, but at the same time they are not dielectrics. One of the differences between semiconductors is that when heated or illuminated, their resistivity does not increase, but decreases. But their main practically applicable property turned out to be one-way conductivity. Due to the uneven distribution of thermal motion energy in a semiconductor crystal, some atoms are ionized. The released electrons cannot be captured by surrounding atoms, because their valence bonds are saturated. These free electrons can move through the metal, creating an electronic conduction current. At the same time, the atom from whose shell an electron has escaped becomes an ion. This ion is neutralized by capturing a neighboring atom. As a result of such chaotic movement, a movement of the place with the missing ion occurs, which is externally visible as the movement of a positive charge. This is called hole conduction current. In an ideal semiconductor crystal, current is created by the movement of equal numbers of free electrons and holes. This type of conductivity is called intrinsic conductivity. As the temperature decreases, the number of free electrons, proportional to the average energy of the atoms, decreases and the semiconductor becomes similar to a dielectric. To improve conductivity, impurities are sometimes added to a semiconductor, which can be donor (increase the number of electrons without increasing the number of holes) and acceptor (increase the number of holes without increasing the number of electrons). Semiconductors where the number of electrons exceeds the number of holes are called electronic semiconductors, or n-type semiconductors. Semiconductors where the number of holes exceeds the number of electrons are called hole semiconductors, or p-type semiconductors.
43. Semiconductor diode. Transistor.
A semiconductor diode consists of p-n transition, i.e. of two connected semiconductors of different conductivity types. When connecting, electrons diffuse into R-semiconductor. This leads to the appearance in the electronic semiconductor of uncompensated positive ions of the donor impurity, and in the hole semiconductor - negative ions of the acceptor impurity that have captured the diffused electrons. An electric field arises between the two layers. If a positive charge is applied to the area with electronic conductivity, and a negative charge to the area with hole conductivity, then the blocking field will increase, the current strength will sharply decrease and is almost independent of voltage. This method of switching on is called blocking, and the current flowing in the diode is called reverse. If a positive charge is applied to the area with hole conductivity, and a negative charge to the area with electron conductivity, then the blocking field will weaken; the current strength through the diode in this case depends only on the resistance of the external circuit. This method of switching is called bypass, and the current flowing in the diode is called direct.
A transistor, also known as a semiconductor triode, consists of two p-n(or n-p) transitions. The middle part of the crystal is called the base, the outer parts are the emitter and collector. Transistors in which the base has hole conductivity are called transistors p-n-p transition. To drive a transistor p-n-p-type voltage of negative polarity relative to the emitter is applied to the collector. The voltage at the base can be either positive or negative. Because there are more holes, then the main current through the junction will be a diffusion flow of holes from R-regions If a small forward voltage is applied to the emitter, then a hole current will flow through it, diffusing from R-regions in n-area (base). But because If the base is narrow, the holes fly through it, accelerated by the field, into the collector. (???, I didn’t understand something here...). The transistor is able to distribute the current, thereby amplifying it. The ratio of the change in current in the collector circuit to the change in current in the base circuit, other things being equal, is a constant value, called the integral transfer coefficient of the base current. Therefore, by changing the current in the base circuit, it is possible to obtain changes in the collector circuit current. (???)
44. Electric current in gases. Types of gas discharges and their application. The concept of plasma.
Gas, when exposed to light or heat, can become a conductor of current. The phenomenon of current passing through a gas under external influence is called a non-self-sustaining electric discharge. The process of formation of gas ions under the influence of temperature is called thermal ionization. The appearance of ions under the influence of light radiation is photoionization. A gas in which a significant portion of the molecules are ionized is called plasma. The plasma temperature reaches several thousand degrees. Plasma electrons and ions are able to move under the influence of an electric field. As the field strength increases, depending on the pressure and nature of the gas, a discharge occurs in it without the influence of external ionizers. This phenomenon is called self-sustained electrical discharge. In order for an electron to ionize an atom when it hits it, it is necessary that it have an energy no less than the ionization work. An electron can acquire this energy under the influence of the forces of an external electric field in a gas along its free path, i.e. 
. Because the mean free path is small, independent discharge is possible only at high field strength. At low gas pressure, a glow discharge is formed, which is explained by an increase in the conductivity of the gas during rarefaction (the free path increases). If the current in a self-discharge is very high, then electron impacts can cause heating of the cathode and anode. At high temperatures, electrons are emitted from the cathode surface, maintaining a discharge in the gas. This type of discharge is called arc.
45. Electric current in a vacuum. Thermionic emission. Cathode-ray tube.
There are no free charge carriers in a vacuum, therefore, without external influence, there is no current in a vacuum. It can occur if one of the electrodes is heated to a high temperature. The heated cathode emits electrons from its surface. The phenomenon of the emission of free electrons from the surface of heated bodies is called thermionic emission. The simplest device using thermionic emission is a vacuum diode. The anode consists of a metal plate, the cathode - of a thin coiled wire. An electron cloud is created around the cathode when it is heated. If you connect the cathode to the positive terminal of the battery and the anode to the negative terminal, then the field inside the diode will bias electrons to the cathode, and no current will flow. If you connect the opposite way - the anode to the plus and the cathode to the minus - then the electric field will move electrons towards the anode. This explains the one-way conductivity property of the diode. The flow of electrons moving from the cathode to the anode can be controlled using an electromagnetic field. To do this, the diode is modified and a grid is added between the anode and cathode. The resulting device is called a triode. If a negative potential is applied to the grid, the field between the grid and the cathode will impede the movement of the electron. If you apply a positive field, the field will impede the movement of electrons. The electrons emitted by the cathode can be accelerated to high speeds using electric fields. The ability of electron beams to be deflected by electromagnetic fields is used in CRTs.
46. Magnetic interaction of currents. A magnetic field. The force acting on a current-carrying conductor in a magnetic field. Magnetic field induction.
If a current of the same direction is passed through the conductors, then they attract, and if they are equal, then they repel. Consequently, there is some interaction between the conductors, which cannot be explained by the presence of an electric field, because In general, conductors are electrically neutral. A magnetic field is created by moving electric charges and affects only moving charges. The magnetic field is a special type of matter and is continuous in space. The passage of electric current through a conductor is accompanied by the generation of a magnetic field, regardless of the medium. The magnetic interaction of conductors is used to determine the magnitude of the current. 1 ampere is the current strength passing through two parallel conductors of ¥ length and small cross-section, located at a distance of 1 meter from each other, at which the magnetic flux causes an interaction force downwards equal to each meter of length. The force with which a magnetic field acts on a current-carrying conductor is called the Ampere force. To characterize the ability of a magnetic field to influence a current-carrying conductor, there is a quantity called magnetic induction. The magnetic induction module is equal to the ratio of the maximum value of the Ampere force acting on a current-carrying conductor to the current strength in the conductor and its length. The direction of the induction vector is determined by the rule of the left hand (conductor in the hand, force in the thumb, induction in the palm). The unit of magnetic induction is tesla, equal to the induction of such a magnetic flux in which a maximum ampere force of 1 newton acts on 1 meter of conductor with a current of 1 ampere. A line at any point of which the magnetic induction vector is directed tangentially is called a magnetic induction line. If at all points of some space the induction vector has the same absolute value and the same direction, then the field in this part is called homogeneous. Depending on the angle of inclination of the current-carrying conductor relative to the magnetic induction vector of the Ampere forces, it changes in proportion to the sine of the angle.
47. Ampere's law. The effect of a magnetic field on a moving charge. Lorentz force.
The effect of a magnetic field on a current in a conductor indicates that it acts on moving charges. Current strength I in a conductor is related to the concentration n free charged particles, speed v their ordered movement and area S cross-section of the conductor by the expression , where q– charge of one particle. Substituting this expression into the Ampere force formula, we get 
. Because nSl equal to the number of free particles in a conductor of length l, then the force acting from the field on one charged particle moving at speed v at an angle a to the magnetic induction vector B equal to 
. This force is called the Lorentz force. The direction of the Lorentz force for a positive charge is determined by the left-hand rule. In a uniform magnetic field, a particle moving perpendicular to the magnetic field induction lines acquires centripetal acceleration under the influence of the Lorentz force 
and moves in a circle. The radius of the circle and the period of revolution are determined by the expressions 
. The independence of the orbital period from radius and speed is used in a charged particle accelerator - a cyclotron.
48. Magnetic properties of matter. Ferromagnets.
Electromagnetic interaction depends on the environment in which the charges are located. If you hang a small one near a large coil, it will deviate. If an iron core is inserted into the larger one, the deviation will increase. This change shows that the induction changes when the core is introduced. Substances that significantly enhance an external magnetic field are called ferromagnets. A physical quantity that shows how many times the inductance of a magnetic field in a medium differs from the inductance of a field in a vacuum is called magnetic permeability. Not all substances enhance a magnetic field. Paramagnets create a weak field that coincides in direction with the external one. Diamagnets weaken the external field with their field. Ferromagnetism is explained by the magnetic properties of the electron. An electron is a moving charge and therefore has its own magnetic field. In some crystals, conditions exist for parallel orientation of the magnetic fields of electrons. As a result, magnetized areas called domains appear inside the ferromagnetic crystal. As the external magnetic field increases, the domains order their orientation. At a certain value of induction, complete ordering of the orientation of the domains occurs and magnetic saturation occurs. When a ferromagnet is removed from an external magnetic field, not all domains lose their orientation, and the body becomes a permanent magnet. The orderly orientation of domains can be disrupted by thermal vibrations of atoms. The temperature at which a substance ceases to be ferromagnetic is called the Curie temperature.
49. Electromagnetic induction. Magnetic flux. Law of electromagnetic induction. Lenz's rule.
In a closed circuit, when the magnetic field changes, an electric current arises. This current is called induced current. The phenomenon of current generation in a closed circuit due to changes in the magnetic field penetrating the circuit is called electromagnetic induction. The appearance of current in a closed circuit indicates the presence of external forces of a non-electrostatic nature or the occurrence of induced emf. A quantitative description of the phenomenon of electromagnetic induction is given on the basis of establishing the connection between the induced emf and magnetic flux. Magnetic flux F through the surface is a physical quantity equal to the product of the surface area S per module of the magnetic induction vector B and by the cosine of the angle a between it and the normal to the surface. The unit of magnetic flux is the weber, which is equal to the flux that, when uniformly decreasing to zero in 1 second, causes an emf of 1 volt. The direction of the induction current depends on whether the flux passing through the circuit increases or decreases, as well as on the direction of the field relative to the circuit. The general formulation of Lenz's rule: the induced current arising in a closed circuit has such a direction that the magnetic flux created by it through the area limited by the circuit tends to compensate for the change in the magnetic flux that causes this current. Law of electromagnetic induction: The induced emf in a closed circuit is directly proportional to the rate of change of the magnetic flux through the surface bounded by this circuit and is equal to the rate of change of this flux, taking into account Lenz's rule. When the EMF changes in a coil consisting of n identical turns, the total emf in n times the emf in one single turn. For a uniform magnetic field, based on the definition of magnetic flux, it follows that the induction is equal to 1 Tesla if the flux through a circuit of 1 square meter is equal to 1 Weber. The occurrence of an electric current in a stationary conductor is not explained by magnetic interaction, because The magnetic field acts only on moving charges. The electric field that arises when the magnetic field changes is called an eddy electric field. The work of the vortex field forces to move charges is the induced emf. The vortex field is not associated with charges and represents closed lines. The work done by the forces of this field along a closed loop can be different from zero. The phenomenon of electromagnetic induction also occurs when the source of magnetic flux is at rest and the conductor is moving. In this case, the cause of the occurrence of an induced emf equal to 
, is the Lorentz force.
50. The phenomenon of self-induction. Inductance. Magnetic field energy.
Electric current passing through a conductor creates a magnetic field around it. Magnetic flux F through the circuit is proportional to the magnetic induction vector IN, and induction, in turn, is the current strength in the conductor. Therefore, for magnetic flux we can write . The proportionality coefficient is called inductance and depends on the properties of the conductor, its size and the environment in which it is located. The unit of inductance is henry, inductance is equal to 1 henry if, at a current strength of 1 ampere, the magnetic flux is equal to 1 weber. When the current in the coil changes, the magnetic flux created by this current changes. A change in magnetic flux causes an induced emf to appear in the coil. The phenomenon of the occurrence of induced emf in a coil as a result of a change in current strength in this circuit is called self-induction. In accordance with Lenz's rule, the self-inductive emf prevents an increase when turning on and a decrease when turning off the circuit. Self-induced emf arising in an inductive coil L, according to the law of electromagnetic induction is equal to 
. Suppose that when the network is disconnected from the source, the current decreases according to a linear law. Then the self-induction emf has a constant value equal to 
. During t with a linear decrease, a charge will pass through the circuit. In this case, the work done by the electric current is equal to 
. This work is done for the light of energy W m magnetic field of the coil.
51. Harmonic vibrations. Amplitude, period, frequency and phase of oscillations.
Mechanical vibrations are movements of bodies that repeat exactly or approximately the same at regular intervals. The forces acting between bodies within the system of bodies under consideration are called internal forces. The forces acting on the bodies of the system from other bodies are called external forces. Free vibrations are vibrations that arise under the influence of internal forces, for example, a pendulum on a string. Vibrations under the influence of external forces are forced oscillations, for example, a piston in an engine. The common feature of all types of vibrations is the repeatability of the movement process after a certain time interval. Harmonic vibrations are those described by the equation 
. In particular, oscillations that occur in a system with one restoring force proportional to the deformation are harmonic. The minimum interval through which the movement of a body is repeated is called the period of oscillation T. A physical quantity that is the inverse of the oscillation period and characterizes the number of oscillations per unit time is called frequency. Frequency is measured in hertz, 1 Hz = 1 s -1. The concept of cyclic frequency is also used, which determines the number of oscillations in 2p seconds. The magnitude of the maximum displacement from the equilibrium position is called the amplitude. The value under the cosine sign is the phase of oscillation, j 0 is the initial phase of oscillation. The derivatives also change harmonically, and , and the total mechanical energy for an arbitrary deviation X(angle, coordinate, etc.) is equal to 
, Where A And IN– constants determined by the system parameters. By differentiating this expression and taking into account the absence of external forces, it is possible to write down that , from where .
52. Mathematical pendulum. Oscillations of a load on a spring. The period of oscillation of a mathematical pendulum and a load on a spring.
A small body suspended on an inextensible thread, the mass of which is negligible compared to the mass of the body, is called a mathematical pendulum. The vertical position is an equilibrium position in which the force of gravity is balanced by the force of elasticity. For small deviations of the pendulum from the equilibrium position, a resultant force appears directed towards the equilibrium position, and its oscillations are harmonic. The period of harmonic oscillations of a mathematical pendulum with a small swing angle is equal to . To derive this formula, let's write down Newton's second law for a pendulum. The pendulum is acted upon by gravity and the tension of the string. Their resultant at a small angle of deflection is equal to . Hence, 
, where 
.
During harmonic vibrations of a body suspended on a spring, the elastic force is equal according to Hooke's law. According to Newton's second law.
53. Energy conversion during harmonic vibrations. Forced vibrations. Resonance.
When a mathematical pendulum deviates from its equilibrium position, its potential energy increases, because the distance to the Earth increases. When moving towards the equilibrium position, the speed of the pendulum increases, and the kinetic energy increases, due to a decrease in the potential reserve. In the equilibrium position, kinetic energy is maximum, potential energy is minimum. In the position of maximum deviation it is the other way around. With a spring it’s the same, but it’s not the potential energy in the Earth’s gravitational field that is taken, but the potential energy of the spring. Free oscillations always turn out to be damped, i.e. with decreasing amplitude, because energy is spent on interaction with surrounding bodies. The energy losses in this case are equal to the work of external forces during the same time. The amplitude depends on the frequency of the force change. It reaches its maximum amplitude when the oscillation frequency of the external force coincides with the natural oscillation frequency of the system. The phenomenon of increasing the amplitude of forced oscillations under the described conditions is called resonance. Since during resonance the external force performs maximum positive work over a period, the resonance condition can be defined as the condition for maximum energy transfer to the system.
54. Propagation of vibrations in elastic media. Transverse and longitudinal waves. Wavelength. Relationship between wavelength and the speed of its propagation. Sound waves. Sound speed. Ultrasound
Excitation of oscillations in one place of the medium causes forced oscillations of neighboring particles. The process of vibrations propagating in space is called a wave. Waves in which vibrations occur perpendicular to the direction of propagation are called transverse waves. Waves in which oscillations occur along the direction of propagation of the wave are called longitudinal waves. Longitudinal waves can arise in all media, transverse waves - in solids under the influence of elastic forces during deformation or surface tension forces and gravity. The speed of propagation of oscillations v in space is called wave speed. The distance l between points closest to each other, oscillating in the same phases, is called the wavelength. The dependence of wavelength on speed and period is expressed as , or . When waves arise, their frequency is determined by the oscillation frequency of the source, and the speed is determined by the medium where they propagate, so waves of the same frequency can have different lengths in different media. The processes of compression and rarefaction in air spread in all directions and are called sound waves. Sound waves are longitudinal. The speed of sound depends, like the speed of any waves, on the medium. In air the speed of sound is 331 m/s, in water – 1500 m/s, in steel – 6000 m/s. Sound pressure is additionally the pressure in a gas or liquid caused by a sound wave. Sound intensity is measured by the energy transferred by sound waves per unit time through a unit cross-sectional area perpendicular to the direction of propagation of the waves, and is measured in watts per square meter. The intensity of a sound determines its volume. The pitch of the sound is determined by the frequency of vibration. Ultrasound and infrasound are sound vibrations that lie beyond the limits of audibility with frequencies of 20 kilohertz and 20 hertz, respectively.
55.Free electromagnetic oscillations in the circuit. Conversion of energy in an oscillatory circuit. Natural frequency of oscillations in the circuit.
An electric oscillatory circuit is a system consisting of a capacitor and a coil connected in a closed circuit. When a coil is connected to a capacitor, a current arises in the coil and the energy of the electric field is converted into magnetic field energy. The capacitor does not discharge instantly, because... this is prevented by the self-induced emf in the coil. When the capacitor is completely discharged, the self-inductive emf will prevent the current from decreasing, and the energy of the magnetic field will be converted into electric energy. The current arising in this case will charge the capacitor, and the sign of the charge on the plates will be opposite to the original one. After which the process is repeated until all the energy is spent on heating the circuit elements. Thus, the energy of the magnetic field in the oscillatory circuit is converted into electric energy and vice versa. For the total energy of the system it is possible to write the following relations: 
, from where for an arbitrary moment of time 
. As is known, for a complete chain 
. Believing that in an ideal case R»0, we finally get , or . The solution to this differential equation is the function 
, Where . The value w is called the natural circular (cyclic) frequency of oscillations in the circuit.
56. Forced electrical oscillations. Alternating electric current. Alternator. AC power.
Alternating current in electrical circuits is the result of the excitation of forced electromagnetic oscillations in them. Let a flat coil have area S and induction vector B makes an angle j with the perpendicular to the plane of the coil. Magnetic flux F in this case, through the area of the turn is determined by the expression . When the coil rotates with a frequency n, the angle j changes according to the law., then the expression for the flow takes the form. Changes in magnetic flux create an induced emf equal to minus the rate of change of flux. Consequently, the change in induced emf will occur according to the harmonic law. The voltage removed from the output of the generator is proportional to the number of turns of the winding. When the voltage changes according to the harmonic law 
The field strength in the conductor changes according to the same law. Under the influence of the field, something appears whose frequency and phase coincide with the frequency and phase of voltage oscillations. Fluctuations in current strength in the circuit are forced, occurring under the influence of applied alternating voltage. When the phases of current and voltage coincide, the alternating current power is equal to or 
. The average value of the squared cosine over the period is 0.5, therefore . The effective value of current is the direct current that releases the same amount of heat in the conductor as alternating current. At amplitude Imax harmonic oscillations of the current, the effective voltage is equal to . The effective voltage value is also several times less than its amplitude value. The average current power when the oscillation phases coincide is determined through the effective voltage and current strength.
5 7. Active, inductive and capacitive reactance.
Active resistance R is a physical quantity equal to the ratio of power to the square of current, which is obtained from the expression for power. At low frequencies it is practically independent of frequency and coincides with the electrical resistance of the conductor.
Let a coil be connected to an alternating current circuit. Then, when the current changes according to the law, a self-inductive emf appears in the coil. Because the electrical resistance of the coil is zero, then the emf is equal to minus the voltage at the ends of the coil created by an external generator (??? What other generator???). Therefore, a change in current causes a change in voltage, but with a phase shift 
. The product is the amplitude of voltage oscillations, i.e. 
. The ratio of the amplitude of voltage oscillations across the coil to the amplitude of current oscillations is called inductive reactance 
.
Let there be a capacitor in the circuit. When it is turned on, it charges for a quarter of the period, then discharges for the same amount, then the same thing, but with a change in polarity. When the voltage across the capacitor changes according to the harmonic law the charge on its plates is equal to . The current in the circuit occurs when the charge changes: , similar to the case with a coil, the amplitude of the current fluctuations is equal to 
. The value equal to the ratio of the amplitude to the current strength is called capacitive reactance 
.
58. Ohm's law for alternating current.
Consider a circuit consisting of a resistor, a coil, and a capacitor connected in series. At any time, the applied voltage is equal to the sum of the voltages on each element. Fluctuations in current strength in all elements occur according to the law. Voltage fluctuations on the resistor coincide in phase with current fluctuations, voltage fluctuations on the capacitor lag behind current fluctuations in phase, voltage fluctuations on the coil lead current fluctuations in phase by (why are they lagging behind???). Therefore, the condition for the sum of stresses to be equal to the total can be written as: Using a vector diagram, you can see that the voltage amplitude in the circuit is equal to , or , i.e. 
. The total resistance of the circuit is denoted by 
. From the diagram it is obvious that the voltage also fluctuates according to the harmonic law . The initial phase j can be found using the formula 
. The instantaneous power in the alternating current circuit is equal. Since the average value of the squared cosine over the period is 0.5, . If there is a coil and a capacitor in the circuit, then according to Ohm's law for alternating current. The value is called power factor.
59. Resonance in an electrical circuit.
Capacitive and inductive reactance depend on the frequency of the applied voltage. Therefore, at a constant voltage amplitude, the amplitude of the current depends on the frequency. At a frequency value at which , the sum of the voltages on the coil and capacitor becomes zero, because their oscillations are opposite in phase. As a result, the voltage across the active resistance at resonance turns out to be equal to the full voltage, and the current reaches its maximum value. Let us express the inductive and capacitive reactance at resonance: 
, hence 
. This expression shows that at resonance, the amplitude of voltage oscillations on the coil and capacitor can exceed the amplitude of oscillations of the applied voltage.
60. Transformer.
A transformer consists of two coils with different numbers of turns. When voltage is applied to one of the coils, a current appears in it. If the voltage changes according to a harmonic law, then the current will change according to the same law. The magnetic flux passing through the coil is equal to 
. When the magnetic flux changes, a self-inductive emf occurs in each turn of the first coil. The product is the amplitude of the emf in one turn, the total emf in the primary coil. The secondary coil is penetrated by the same magnetic flux, therefore . Because magnetic fluxes are the same, then. The active resistance of the winding is small compared to the inductive resistance, so the voltage is approximately equal to the emf. From here. Coefficient TO called the transformation ratio. Heating losses of wires and cores are small, therefore F1" Ф 2. The magnetic flux is proportional to the current in the winding and the number of turns. Hence, i.e. . Those. transformer increases voltage TO times, reducing the current strength by the same amount. The current power in both circuits, neglecting losses, is the same.
61. Electromagnetic waves. The speed of their spread. Properties of electromagnetic waves.
Any change in the magnetic flux in the circuit causes an induction current to appear in it. Its appearance is explained by the emergence of a vortex electric field with any change in the magnetic field. A vortex electric hearth has the same property as an ordinary one - to generate a magnetic field. Thus, once the process of mutual generation of magnetic and electric fields has begun, it continues continuously. Electric and magnetic fields that make up electromagnetic waves can exist in a vacuum, unlike other wave processes. From experiments with interference, the speed of propagation of electromagnetic waves was established to be approximately . In the general case, the speed of an electromagnetic wave in an arbitrary medium is calculated by the formula. The energy densities of the electric and magnetic components are equal to each other: 
, where . The properties of electromagnetic waves are similar to the properties of other wave processes. When passing through the interface between two media, they are partially reflected and partially refracted. They are not reflected from the dielectric surface; they are reflected almost completely from metals. Electromagnetic waves have the properties of interference (Hertz's experiment), diffraction (aluminum plate), polarization (mesh).
62. Principles of radio communication. The simplest radio receiver.
To carry out radio communication, it is necessary to ensure the possibility of emitting electromagnetic waves. The greater the angle between the capacitor plates, the more freely EM waves propagate in space. In reality, an open circuit consists of a coil and a long wire - an antenna. One end of the antenna is grounded, the other is raised above the Earth's surface. Because Since the energy of electromagnetic waves is proportional to the fourth power of frequency, EM waves practically do not arise when alternating current oscillates at sound frequencies. Therefore, the principle of modulation is used - frequency, amplitude or phase. The simplest generator of modulated oscillations is shown in the figure. Let the oscillation frequency of the circuit change according to the law. Let the frequency of modulated sound vibrations also change as 
, and W<
.
Let's consider a simple radio receiver. It consists of an antenna, an oscillating circuit with a variable capacitor, a detector diode, a resistor and a telephone. The frequency of the oscillatory circuit is selected so that it coincides with the carrier frequency, and the amplitude of oscillations on the capacitor becomes maximum. This allows you to select the desired frequency from all received ones. From the circuit, modulated high-frequency oscillations enter the detector. After passing the detector, the current charges the capacitor every half cycle, and the next half cycle, when the current does not pass through the diode, the capacitor is discharged through the resistor. (Did I understand correctly???).
64. Analogy between mechanical and electrical vibrations.
The analogies between mechanical and electrical vibrations look like this:
Coordinate | |||
Speed | Current strength | ||
Acceleration | Rate of change of current | ||
Inductance | |||
Rigidity | Reciprocal value electrical capacity | ||
Voltage | |||
Viscosity | Resistance | ||
Potential energy deformed spring | Electric field energy capacitor | ||
Kinetic energy, where . 65. Electromagnetic radiation scale. Dependence of the properties of electromagnetic radiation on frequency. Application of electromagnetic radiation. The range of electromagnetic waves with a length from 10 -6 m to m are radio waves. Used for television and radio communications. Lengths from 10 -6 m to 780 nm - infrared waves. Visible light – from 780 nm to 400 nm. Ultraviolet radiation – from 400 to 10 nm. Radiation in the range from 10 nm to 10 pm is X-ray radiation. Gamma radiation corresponds to shorter wavelengths. (Application???). The shorter the wavelength (hence, the higher the frequency), the less waves are absorbed by the medium. 65. Rectilinear propagation of light. Speed of light. Laws of reflection and refraction of light. The straight line indicating the direction of propagation of light is called a light ray. At the boundary of two media, light can be partially reflected and propagate in the first medium in a new direction, and also partially pass through the boundary and propagate in the second medium. The incident beam, the reflected beam, and the beam perpendicular to the boundary of the two media, reconstructed at the point of incidence, lie in the same plane. The angle of reflection is equal to the angle of incidence. This law coincides with the law of reflection of waves of any nature and is proven by Huygens' principle. When light passes through the interface between two media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for the two given media.<рисунок>. Magnitude n called the refractive index. The refractive index of a medium relative to vacuum is called the absolute refractive index of that medium. When observing the effect of refraction, it can be noted that in the case of a transition of a medium from an optically denser medium to a less dense one, with a gradual increase in the angle of incidence, it can be reached such a value that the angle of refraction becomes equal to . In this case the equality is satisfied. The angle of incidence a 0 is called the limiting angle of total reflection. At angles greater than a 0, total reflection occurs. 66. Lens, image construction. Lens formula. A lens is a transparent body bounded by two spherical surfaces. A lens that is thicker at the edges than in the middle is called concave, while a lens that is thicker in the middle is called convex. The straight line passing through the centers of both spherical surfaces of the lens is called the main optical axis of the lens. If the thickness of the lens is small, then the main optical axis can be said to intersect with the lens at one point, called the optical center of the lens. The straight line passing through the optical center is called the secondary optical axis. If a beam of light parallel to the main optical axis is directed at a lens, then at a convex lens the beam will converge at a point F. In the lens formula, the distance from the lens to the virtual image is considered negative. The optical power of a biconvex (and indeed any) lens is determined from the radius of its curvature and the refractive index of glass and air 66. Coherence. Interference of light and its application in technology. Diffraction of light. Diffraction grating. The wave properties of light are observed in the phenomena of diffraction and interference. Two light frequencies whose phase difference is zero are said to be coherent with each other. During interference - the addition of coherent waves - an interference pattern of maxima and minima of illumination that is stable over time appears. With a path difference, an interference maximum occurs, at 67. Dispersion of light. Spectrum of electromagnetic radiation. Spectroscopy. Spectral analysis. Sources of radiation and types of spectra. A narrow parallel beam of white light, when passing through a prism, is decomposed into beams of light of different colors. The color band visible in this case is called the continuous spectrum. The phenomenon of the dependence of the speed of light on the wavelength (frequency) is called light dispersion. This effect is explained by the fact that white light consists of EM waves of different wavelengths, on which the refractive index depends. It has the greatest value for the shortest wave - violet, and the least - for red. In a vacuum, the speed of light is the same regardless of its frequency. If the source of the spectrum is a rarefied gas, then the spectrum looks like narrow lines on a black background. Compressed gases, liquids and solids emit a continuous spectrum, where colors smoothly blend into each other. The nature of the spectrum is explained by the fact that each element has its own specific set of emitted spectrum. This property allows the use of spectral analysis to determine the chemical composition of a substance. A spectroscope is a device that is used to study the spectral composition of light emitted by a certain source. Decomposition is carried out using a diffraction grating (better) or a prism; quartz optics is used to study the ultraviolet region. 68. Photoelectric effect and its laws. Quanta of light. Einstein's equation for the photoelectric effect. Application of the photoelectric effect in technology. The phenomenon of electrons being ejected from solids and liquids under the influence of light is called the external photoelectric effect, and electrons ejected in this way are called photoelectrons. The laws of the photoelectric effect have been established experimentally - the maximum speed of photoelectrons is determined by the frequency of light and does not depend on its intensity; for each substance there is its own red limit of the photoelectric effect, i.e. such a frequency n min at which the photoelectric effect is still possible, the number of photoelectrons ejected per second is directly proportional to the light intensity. The inertia-free photoelectric effect has also been established - it occurs instantly after the start of illumination, provided the red limit is exceeded. The photoelectric effect can be explained using quantum theory, which asserts the discreteness of energy. An electromagnetic wave, according to this theory, consists of separate portions - quanta (photons). When a quantum of energy is absorbed, the photoelectron acquires kinetic energy, which can be found from Einstein's equation for the photoelectric effect 69. Rutherford's experiments on the scattering of alpha particles. Nuclear model of the atom. Bohr's quantum postulates. The first model of the structure of the atom belongs to Thomson. He suggested that an atom is a positively charged ball, inside of which there are inclusions of negatively charged electrons. Rutherford conducted an experiment on implanting fast alpha particles into a metal plate. At the same time, it was observed that some of them deviate slightly from rectilinear propagation, and some - at angles greater than 2 0 . This was explained by the fact that the positive charge in the atom is not contained uniformly, but in a certain volume, much smaller than the size of the atom. This central part was called the nucleus of the atom, where the positive charge and almost all the mass are concentrated. The radius of the atomic nucleus has dimensions of the order of 10 -15 m. Rutherford also proposed the so-called. planetary model of the atom, according to which electrons revolve around the atom like planets around the Sun. Radius of the farthest orbit = radius of the atom. But this model contradicted electrodynamics, because accelerated movement (including electrons in a circle) is accompanied by the emission of EM waves. Consequently, the electron gradually loses its energy and must fall onto the nucleus. In reality, neither radiation nor falling of the electron occurs. An explanation for this was given by N. Bohr, putting forward two postulates - an atomic system can only be in certain specific states in which there is no emission of light, although the movement is accelerated, and when transitioning from one state to another, either absorption or emission of a quantum occurs according to the law , where is Planck's constant. The various possible stationary states are determined from the relation 70. Emission and absorption of light by atoms. Laser. Atoms can spontaneously emit quanta of light, while it passes incoherently (since each atom emits independently of the others) and is called spontaneous. The transition of an electron from an upper level to a lower one can occur under the influence of an external electromagnetic field with a frequency equal to the transition frequency. Such radiation is called forced (induced). Those. As a result of the interaction of an excited atom with a photon of the corresponding frequency, the probability of the appearance of two identical photons with the same direction and frequency is high. The peculiarity of stimulated emission is that it is monochromatic and coherent. This property is the basis for the operation of lasers (optical quantum generators). In order for a substance to amplify light passing through it, more than half of its electrons must be in an excited state. This state is called a state with inverted population of levels. In this case, absorption of photons will occur less frequently than emission. To operate a laser on a ruby rod, the so-called. a pumping lamp, the purpose of which is to create a population inversion. Moreover, if one atom moves from the metastable state to the ground state, a chain reaction of photon emission will occur. With the appropriate (parabolic) shape of the reflecting mirror, it is possible to create a beam in one direction. Complete illumination of all excited atoms occurs in 10 -10 s, so the laser power reaches billions of watts. There are also lasers using gas lamps, the advantage of which is the continuity of radiation. 70. Composition of the nucleus of an atom. Isotopes. Binding energy of atomic nuclei. Nuclear reactions. Electric charge of an atom nucleus q equal to the product of the elementary electric charge e per serial number Z chemical element in the periodic table. Atoms that have the same structure have the same electron shell and are chemically indistinguishable. Nuclear physics uses its own units of measurement. 1 Fermi – 1 femtometer, . 1 atomic mass unit is 1/12 the mass of a carbon atom. . Atoms with the same nuclear charge but different masses are called isotopes. Isotopes differ in their spectra. The nucleus of an atom consists of protons and neutrons. The number of protons in the nucleus is equal to the charge number Z, number of neutrons – mass minus number of protons A–Z=N. The positive charge of a proton is numerically equal to the charge of an electron, the mass of a proton is 1.007 amu. The neutron has no charge and has a mass of 1.009 amu. (a neutron is more than two electron masses heavier than a proton). Neutrons are stable only in the composition of atomic nuclei; in their free form, they live for ~15 minutes and decay into a proton, electron and antineutrino. The force of gravitational attraction between nucleons in the nucleus exceeds the electrostatic repulsive force by 10 36 times. The stability of nuclei is explained by the presence of special nuclear forces. At a distance of 1 fm from the proton, nuclear forces are 35 times higher than Coulomb forces, but they decrease very quickly, and at a distance of about 1.5 fm they can be neglected. Nuclear forces do not depend on whether the particle has a charge. Accurate measurements of the masses of atomic nuclei have shown the existence of a difference between the mass of a nucleus and the algebraic sum of the masses of its constituent nucleons. To separate an atomic nucleus into its components, energy must be expended. The quantity is called mass defect. The minimum energy that must be expended to separate a nucleus into its constituent nucleons is called the binding energy of the nucleus, which is spent on doing work against nuclear attractive forces. The ratio of binding energy to mass number is called specific binding energy. A nuclear reaction is the transformation of the original atomic nucleus upon interaction with any particle into another, different from the original one. As a result of a nuclear reaction, particles or gamma rays can be emitted. There are two types of nuclear reactions: some require the expenditure of energy, while others release energy. The released energy is called the output of a nuclear reaction. In nuclear reactions, all conservation laws are satisfied. The law of conservation of angular momentum takes the form of the law of conservation of spin. 71. Radioactivity. Types of radioactive radiation and their properties. Nuclei have the ability to spontaneously decay. In this case, only those nuclei that have minimal energy are stable compared to those into which the nucleus can spontaneously transform. Nuclei in which there are more protons than neutrons are unstable because the Coulomb repulsion force increases. Nuclei with more neutrons are also unstable, because The mass of a neutron is greater than the mass of a proton, and an increase in mass leads to an increase in energy. Nuclei can be released from excess energy either by dividing into more stable parts (alpha decay and fission) or by changing their charge (beta decay). Alpha decay is the spontaneous division of an atomic nucleus into an alpha particle and a product nucleus. All elements heavier than uranium are subject to alpha decay. The ability of an alpha particle to overcome the attraction of the nucleus is determined by the tunnel effect (Schrodinger equation). During alpha decay, not all the energy of the nucleus is converted into the kinetic energy of motion of the product nucleus and alpha particle. Part of the energy can be used to excite the product nucleus atom. Thus, some time after decay, the core of the product emits several gamma quanta and returns to its normal state. There is also another type of decay - spontaneous nuclear fission. The lightest element capable of such decay is uranium. Decay occurs according to the law where T– half-life, a constant for a given isotope. Beta decay is a spontaneous transformation of an atomic nucleus, as a result of which its charge increases by one due to the emission of an electron. But the mass of a neutron exceeds the sum of the masses of a proton and an electron. This is explained by the release of another particle - the electron antineutrino. 72. Methods for recording ionizing radiation. The photoemulsion method involves applying a sample to a photographic plate, and after developing it, based on the thickness and length of the particle trace on it, it is possible to determine the amount and distribution of a particular radioactive substance in the sample. A scintillation counter is a device in which one can observe the transformation of the kinetic energy of a fast particle into the energy of a light flash, which, in turn, initiates a photoelectric effect (electric current pulse), which is amplified and recorded. A cloud chamber is a glass chamber filled with air and supersaturated alcohol vapor. As a particle moves through the chamber, it ionizes molecules around which condensation immediately begins. The chain of droplets formed as a result forms a particle track. The bubble chamber works on the same principles, but the recorder is a liquid close to the boiling point. Gas-discharge counter (Geiger counter) is a cylinder filled with rarefied gas and a stretched thread of conductor. The particle causes ionization of the gas; the ions, under the influence of an electric field, diverge to the cathode and anode, ionizing other atoms along the way. A corona discharge occurs, the pulse of which is recorded. 73. Chain reaction of fission of uranium nuclei. In the 30s, it was experimentally established that when uranium is irradiated with neutrons, lanthanum nuclei are formed, which could not be formed as a result of alpha or beta decay. The uranium-238 nucleus consists of 82 protons and 146 neutrons. When dividing exactly in half, praseodymium should be formed, but in a stable praseodymium nucleus there are 9 fewer neutrons. Therefore, when uranium fissions, other nuclei and an excess of free neutrons are formed. In 1939, the first artificial fission of a uranium nucleus was carried out. In this case, 2-3 free neutrons and 200 MeV of energy were released, and about 165 MeV was released in the form of kinetic energy of fragment nuclei or or. Under favorable conditions, the released neutrons can cause fission of other uranium nuclei. The neutron multiplication factor characterizes how the reaction will proceed. If it is more than one. then with each division the number of neutrons increases, the uranium heats up to a temperature of several million degrees, and a nuclear explosion occurs. When the fission coefficient is less than one, the reaction decays, and when it is equal to one, it is maintained at a constant level, which is used in nuclear reactors. Of the natural isotopes of uranium, only the nucleus is capable of fission, and the most common isotope absorbs a neutron and turns into plutonium according to the scheme. Plutonium-239 is similar in properties to uranium-235. 74. Nuclear reactor. Thermonuclear reaction. There are two types of nuclear reactors - slow and fast neutrons. Most of the neutrons released during fission have an energy of the order of 1-2 MeV, and a speed of about 10 7 m/s. Such neutrons are called fast, and are absorbed equally effectively by both uranium-235 and uranium-238, and since There is more heavy isotope, but it does not divide, then the chain reaction does not develop. Neutrons moving at speeds of about 2×10 3 m/s are called thermal. Such neutrons are absorbed by uranium-235 more actively than fast ones. Thus, to carry out a controlled nuclear reaction, it is necessary to slow down the neutrons to thermal speeds. The most common moderators in reactors are graphite, ordinary and heavy water. To ensure that the division coefficient is maintained at unity, absorbers and reflectors are used. The absorbers are rods made of cadmium and boron, which capture thermal neutrons, and the reflector is beryllium. If uranium enriched with an isotope with a mass of 235 is used as fuel, then the reactor can operate without a moderator using fast neutrons. In such a reactor, most of the neutrons are absorbed by uranium-238, which through two beta decays becomes plutonium-239, also a nuclear fuel and the starting material for nuclear weapons. Thus, a fast neutron reactor is not only a power plant, but also a fuel multiplier for the reactor. The disadvantage is the need to enrich uranium with a light isotope. Energy in nuclear reactions is released not only due to the fission of heavy nuclei, but also due to the combination of light ones. To connect nuclei, it is necessary to overcome the Coulomb repulsive force, which is possible at a plasma temperature of about 10 7 –10 8 K. An example of a thermonuclear reaction is the synthesis of helium from deuterium and tritium or 75. Biological effects of ionizing radiation. Radiation protection. Application of radioactive isotopes. A measure of the impact of any type of radiation on a substance is the absorbed dose of radiation. The dose unit is the gray, equal to the dose to which 1 joule of energy is transferred to an irradiated substance weighing 1 kg. Because Since the physical effect of any radiation on a substance is associated not so much with heating, but with ionization, a unit of exposure dose has been introduced, which characterizes the ionization effect of radiation on air. The non-systemic unit of exposure dose is the roentgen, equal to 2.58×10 -4 C/kg. With an exposure dose of 1 roentgen, 1 cm 3 of air contains 2 billion ion pairs. With the same absorbed dose, the effect of different types of radiation is different. The heavier the particle, the stronger its effect (however, the heavier it is, the easier it is to hold). The difference in the biological effect of radiation is characterized by a biological effectiveness coefficient equal to unity for gamma rays, 3 for thermal neutrons, 10 for neutrons with an energy of 0.5 MeV. The dose multiplied by the coefficient characterizes the biological effect of the dose and is called the equivalent dose, measured in sieverts. The main mechanism of action on the body is ionization. The ions enter into a chemical reaction with the cell and disrupt its activity, which leads to cell death or mutation. Natural background radiation averages 2 mSv per year, for cities an additional +1 mSv per year. 76. Absoluteness of the speed of light. Service station elements. Relativistic dynamics. It was experimentally established that the speed of light does not depend on the reference system in which the observer is located. It is also impossible to accelerate any elementary particle, such as an electron, to a speed equal to the speed of light. The contradiction between this fact and Galileo's principle of relativity was resolved by A. Einstein. The basis of his [special] theory of relativity was two postulates: any physical processes proceed identically in different inertial frames of reference, the speed of light in a vacuum does not depend on the speed of the light source and the observer. The phenomena described by the theory of relativity are called relativistic. The theory of relativity introduces two classes of particles - those that move at speeds less than With, and with which the reference system can be associated, and those that move with speeds equal With, with which reference systems cannot be associated. Multiplying this inequality () by , we get . This expression represents the relativistic law of addition of velocities, coinciding with Newton’s at v< Own time, i.e. that which acts in the reference frame associated with the particle is invariant, i.e. does not depend on the choice of inertial reference frame. The principle of relativity modifies this statement, saying that in each inertial frame of reference time flows the same way, but there is no single absolute time for all. Coordinate time is related to proper time by the law 77. Elementary particles. Initially, the proton, neutron and electron were classified as elementary particles, and later the photon. When the decay of the neutron was discovered, muons and pions were added to the number of elementary particles. Their mass ranged from 200 to 300 electron masses. Despite the fact that the neutron decays into a channel, an electron and a neutrino, there are no these particles inside it, and it is considered an elementary particle. Most elementary particles are unstable and have half-lives of the order of 10 -6 –10 -16 s. In the relativistic theory of electron motion in an atom developed by Dirac, it followed that an electron could have a twin with an opposite charge. This particle, detected in cosmic rays, is called a positron. Subsequently, it was proven that all particles have their own antiparticles, differing in spin and (if any) charge. There are also true neutral particles that completely coincide with their antiparticles (pi-null meson and eta-null meson). The phenomenon of annihilation is the mutual annihilation of two antiparticles with the release of energy, for example In total, there are 4 types of fundamental (irreducible to others) interactions - gravitational, electromagnetic, weak and strong. Electromagnetic interaction is explained by the exchange of virtual photons (From the Heisenberg uncertainty it follows that in a short time an electron, due to its internal energy, can release a quantum and compensate for the loss of energy by capturing the same one. The emitted quantum is absorbed by another, thus ensuring interaction.), strong - by the exchange of gluons (spin 1, mass 0, carry “color” quark charge), weak – vector bosons. The gravitational interaction is not explained, but the quanta of the gravitational field should theoretically have mass 0, spin 2 (???).
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– minimum. The phenomenon of light deviation from linear propagation when passing the edge of an obstacle is called diffraction of light. This phenomenon is explained by the Huygens-Fresnel principle: a disturbance at any point is the result of the interference of secondary waves emitted by each element of the wave surface. Diffraction is used in spectral instruments. The element of these devices is a diffraction grating, which is a transparent plate coated with a system of opaque parallel stripes located at a distance d from each other. let a monochromatic wave fall on the grating. As a result of diffraction, light from each slit propagates not only in the original direction, but also in all others. If you place a lens behind the grating, then in the focal plane the parallel rays from all the slits will be collected into one strip. Parallel rays travel with a path difference. When the path difference is equal to an integer number of waves, an interference maximum of light is observed. For each wavelength, the maximum condition is satisfied at its own angle j, so the grating decomposes white light into a spectrum. The longer the wavelength, the larger the angle.
, where A 0 is the work function, a parameter of the substance. The number of photoelectrons leaving the metal surface is proportional to the number of electrons, which, in turn, depends on the illumination (light intensity).
, Where n– an integer. For the motion of an electron in a circle in a hydrogen atom, the following expression is valid: the Coulomb force of interaction with the nucleus. From here. Those. in view of Bohr's postulate about the quantization of energy, movement is possible only in stationary circular orbits, the radii of which are defined as . All states, except one, are conditionally stationary, and only in one - the ground state, in which the electron has a minimum amount of energy - can the atom remain for as long as desired, and the remaining states are called excited.
. Not only the neutron can decay. A free proton is stable, but when exposed to particles it can decay into a neutron, positron and neutrino. If the energy of the new nucleus is less, then positron beta decay occurs 
. Like alpha decay, beta decay can also be accompanied by gamma radiation.
. The synthesis of 1 gram of helium releases energy equivalent to burning 10 tons of diesel fuel. A controlled thermonuclear reaction is possible by heating it to the appropriate temperature by passing an electric current through it or using a laser.
. By squaring this expression, we get . Size s called an interval. A consequence of the relativistic law of addition of velocities is the Doppler effect, which characterizes the change in the frequency of oscillations depending on the velocities of the wave source and the observer. When the observer moves at an angle Q to the source, the frequency changes according to the law 
. As you move away from the source, the spectrum shifts to lower frequencies corresponding to a longer wavelength, i.e. towards red, when approaching – towards purple. The momentum also changes at speeds close to With:
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. According to the law of conservation of energy, the released energy is proportional to the sum of the masses of the annihilated particles. According to conservation laws, particles never arise alone. Particles are divided into groups, according to increasing mass - photon, leptons, mesons, baryons.