Consider the motion of a body in a circle. The speed at which a body moves in a circle, called linear speed . It is found by the formula
Let us find out what is the relationship between linear and angular quantities when a body moves in a circle. Linear quantities are path, speed, tangential and normal acceleration, and angular quantities are the angle of rotation, angular velocity and angular acceleration.
Let's find the connection between angular and linear speed. From geometry it is known that the arc length l the central angle is equal to the product of the angle , measured in radians, and the radius of the circle R, i.e. l =R. Let's differentiate this expression with respect to time: 
(R is taken out of the sign of the derivative, since it is constant). But then we get that
= R. (8)
Let us differentiate expression (8) with respect to time 
Noaangular acceleration modulus. That's why
a = R. (9)
Substituting expression (7) into formula (4), we obtain for the normal acceleration modulus
a n = R. (10)
Thus, when moving material point along a circle, both linear and angular quantities can be used to describe its movement. However, when rotating a rigid body, it is convenient to use angular quantities rather than linear ones, since the equations of motion of different points, expressed in angular quantities, are the same for all points of the body, while when using linear quantities they are different.
Rigid body kinematics
Until now, the movement of bodies that could be considered as material points has been studied. Let us now consider the motion of extended bodies. In this case, we will consider the bodies to be absolutely solid (solid). Under hard In mechanics, a body is understood as a body, the relative arrangement of its parts under the conditions of a given problem is considered unchanged.
There are two types of motion of a rigid body: translational and rotational. Progressive called a movement in which a straight line connecting any two points of a body moves in space parallel to itself. At rotational movement all points of the body move in circles, the centers of which lie on one straight line, called axis of rotation . Any complex movement can be represented as the result of the addition of translational and rotational movements.
Let's consider forward motion. During this movement, all points of the body travel the same paths. Therefore they have the same speeds and accelerations. It follows that to describe such a motion of a body, it is enough to select an arbitrary point on it and use the formulas of the kinematics of a material point. Usually its center of gravity is chosen.
During rotational movement different points solid bodies pass different ways and therefore have at different speeds and accelerations. As a result, to characterize such a movement, it is necessary to choose such quantities that will be the same in this moment time for all points of the body. They are the angle of rotation, angular velocity and angular acceleration.
Dynamics of translational motion
From the first lecture it is clear that kinematics describes movement and does not consider the reasons that cause it. However, this question is important from a practical point of view. Dynamics is the study of the relationship between motion and forces acting in a mechanical system. The basis of dynamics is Newton's three laws, which are a generalization of a large number of experimental data. Before moving on to their consideration, let us introduce the concepts of force and body mass.
FORCE.
In everyday life, we constantly have to deal with various interactions. For example, with the attraction of bodies to the Earth, the repulsion and attraction of magnets and currents flowing through wires, the deflection of electron beams in cathode ray tubes when exposed to electric and magnetic fields, etc. To characterize the interaction of bodies, the concept of force is introduced. In mechanics, the force acting on a body is a measure of its interaction with surrounding bodies. The action of force is manifested in the deformation of the body or in its acquisition of acceleration. Force is a vector. Therefore, it is characterized by module, direction and point of application.
WEIGHT
As follows from experience, bodies have the ability to resist changes in the speed they possess, i.e. they counteract the acquisition of acceleration. This property of bodies was called inertia . To characterize the inert properties of bodies, a physical quantity called mass . The greater the mass of the body, the more inert it is. In addition, due to gravitational forces all bodies attract each other. The modulus of these forces depends on the mass of the bodies. Thus, mass also characterizes the gravitational properties of bodies. The larger it is, the greater the force of their gravitational attraction. So, weight- this is a measure of the inertia of bodies during translational motion and a measure of their gravitational interaction.
In SI units, mass is measured in kilograms (kg).
« Physics - 10th grade"
Angular velocity.
Each point of a body rotating around a fixed axis passing through point O moves in a circle, and various points pass in time Δt different ways. So, AA 1 > BB 1 (Fig. 1.62), therefore the modulus of the velocity of point A is greater than the modulus of the velocity of point B. But the radius vectors that determine the position of points A and B rotate during the time Δt by the same angle Δφ.
Angle φ is the angle between the OX axis and the radius vector that determines the position of point A (see Fig. 1.62).
Let the body rotate uniformly, i.e., for any equal periods of time, the radius vectors rotate by equal angles.
The greater the angle of rotation of the radius vector, which determines the position of some point of a rigid body, over a certain period of time, the faster the body rotates and the greater its angular velocity.
Angular velocity of a body during uniform rotation is a quantity equal to the ratio of the angle of rotation of the body υφ to the time period υt during which this rotation occurred.
We will denote angular velocity by the Greek letter ω (omega). Then by definition
Angular velocity in SI is expressed in radians per second (rad/s). For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of the grinding disk is about 140 rad/s.
Angular velocity can be related to rotational speed.
Rotation frequency- number full revolutions per unit of time (in SI for 1 s).
If the body does ν ( greek letter“nu”) revolutions in 1 s, then the time of one revolution is 1/ν seconds.
The time it takes a body to complete one complete revolution is called rotation period and is designated by the letter T.
If φ 0 ≠ 0, then φ - φ 0 = ωt, or φ = φ 0 ± ωt.
Radian is equal to central corner, resting on an arc whose length is equal to the radius of the circle, 1 rad = 57°17"48". Radian angle equal to the ratio the length of the arc of a circle to its radius: φ = l/R.
Angular velocity takes positive values, if the angle between the radius vector, which determines the position of one of the points of the solid body, and the OX axis increases (Fig. 1.63, a), and negative when it decreases (Fig. 1.63, b).
Thus, we can find the position of the points of a rotating body at any time.
Relationship between linear and angular velocities.
The speed of a point moving in a circle is often called linear speed, to emphasize its difference from angular velocity.
We have already noted that when an absolutely rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
Let us establish a connection between the linear speed of any point of a rotating body and its angular speed. A point lying on a circle of radius R will travel a distance of 2πR in one revolution. Since the time of one revolution of the body is the period T, the module of the linear velocity of a point can be found as follows:
Since ω = 2πν, then
The modulus of centripetal acceleration of a point of a body moving uniformly around a circle can be expressed in terms of the angular velocity of the body and the radius of the circle:
Hence,
and cs = ω 2 R.
Let's write down all possible calculation formulas for centripetal acceleration:
We examined the two simplest movements of an absolutely rigid body - translational and rotational. However, any complex motion of an absolutely rigid body can be represented as the sum of two independent motions: translational and rotational.
Based on the law of independence of motion, it is possible to describe the complex motion of an absolutely rigid body.
6.1. How long will it take a wheel with an angular velocity rad/s to make 100 revolutions?
6.2. What is the linear speed of the points earth's surface at latitude 60 0 at daily rotation Earth? The radius of the Earth is taken to be 6400 km.
6.3. When the radius of a circular orbit increases by 4 times artificial satellite Earth, its period of circulation increases 8 times. How many times does the speed of the satellite's orbit change?
6.4 The minute hand of a clock is 3 times longer than the second hand. Find the ratio of the linear velocities of the ends of the arrows.
6.5. The radius of the well gate handle is 3 times greater than radius shaft on which the cable is wound. What is the linear speed of the end of the handle when lifting a bucket from a depth of 10 m in 20 s?
6.6. What distance will the cyclist travel at 60 revolutions of the pedals, if the diameter of the wheel is 70 cm, the driving gear has 48 teeth, and the driven gear has 18 teeth?
6.7. A wheel of radius R rolls along horizontal surface without sliding with angular velocity. What is the speed of the wheel axis, top point, bottom point of the wheel relative to a horizontal surface.
6.8. The modulus of the linear velocity of a point lying on the wheel rim is 2.5 times greater than the modulus of the linear velocity of a point lying 0.03 m closer to the wheel axis. Find the radius of the wheel.
6.9 When a wheel rolls, it often happens that the lower spokes are clearly visible, but the upper spokes seem to merge. Why is that?
6.10 Length minute hand tower clock MGU is equal to 4.5 m. Determine the linear speed of the end of the arrow and the angular speed of the arrow.
6.11 Determine the acceleration of points on the earth's surface at different latitudes due to participation in the daily rotation of the Earth.
6.12. The linear velocity vector (V = 2 m/s) of a point uniformly rotating in a circle rotated by 30 0 in 0.5 s. Find the acceleration of this point.
6.13. A thread with a load suspended on it is wound from a block with a radius of 20 cm. Acceleration of the load is 2 cm/s 2. Determine the angular velocity of the block when the load passes a distance of 100 cm from the initial position. Determine the magnitude and direction of the acceleration of the bottom point of the block at this moment in time.
6.14. The projectile flew out with speed v 0 at an angle to the horizontal. Determine the radius of curvature, normal and tangential acceleration projectile at the top point of the trajectory.
6.15. A material point moves along a circular path of radius 10 cm in accordance with the equation for the path S = t + 2.5t 2. Find the total acceleration in the 2nd second of motion.
6.16. The projectile flies out at an angle of 45 0 to the horizontal. What is the projectile's flight range if the radius of curvature of the trajectory at the point of maximum ascent is 15 km?
6.17. A spherical tank standing on the ground has a radius R. At what is the lowest speed that a stone thrown from the surface of the earth can fly over the tank and touch its top? At what angle to the horizon should the stone be thrown?
6.18. The entrance to one of the highest bridges in Japan has the shape of a helical line wrapping around a cylinder of radius r. The road surface makes an angle with the horizontal plane. Find the acceleration modulus of a car moving along the entrance with a constant absolute speed v.
6.19. The point begins to move uniformly accelerated in a circle with a radius of 1 m and covers a distance of 50 m in 10 s. What is equal to normal acceleration points 8 s after the start of movement?
6.20. The car is moving at a speed v= 60 km/h. How many revolutions per second do its wheels make if they roll along a highway without slipping and the outer diameter of the tires is d = 60 cm?
6.21. A circle of radius 2 m rotates around a fixed axis so that its angle of rotation depends on time according to the law. Find the linear velocity of various points on the circle and the angular acceleration.
6.22. A wheel of radius 0.1 m rotates around a fixed axis so that its angle of rotation depends on time according to the law. Find the average value of angular velocity for the period of time from t=0 to stop. Find the angular and linear speed, as well as the normal, tangential and total acceleration of the points of the wheel rim at the moments of 10 s and 40 s.
6.23. Using the condition of problem 6.7, determine the magnitude and direction of the velocity and acceleration vectors for two points on the wheel rim located at a given moment in time at opposite ends of the horizontal diameter of the wheels.
6.24. The rigid body rotates with angular velocity, where a = 0.5 rad/s 2 and b = 0.06 rad/s 2. Find the modules of angular velocity and angular acceleration at the moment of time t=10 s, as well as the angle between the vectors of angular acceleration and angular velocity at this moment of time.
6.25. A ball of radius R begins to roll without sliding along inclined plane so that its center moves with constant acceleration(Fig. 12). Find, t seconds after the start of movement, the speed and acceleration of points A, B and O.
DYNAMICS OF A MATERIAL POINT
Task
On a cord across fixed block, loads of masses 0.3 and 0.2 kg are placed. At what acceleration is the system moving? What is the tension in the cord while moving?
We use the above procedure for solving problems on dynamics.
1. Let's make a drawing and arrange the forces acting on each body based on its interactions with other bodies.
 |
A body of mass m 1 interacts with the Earth and the thread; it is acted upon by gravity and the tension of the thread. A body of mass m2 also interacts with the Earth and with the thread; it is acted upon by gravity and the tension of the thread.
2. We choose the direction of movement for each body independently. Since we have arranged all the forces acting on each body, we can now consider their movement independently of each other along their direction of motion.
3. We write down the equation of motion (Newton’s 2nd law) for each body:
4. We design these vector equations to selected directions of movement:
F H – F t1 = m 1 a
F H – Ft 2 = m 2 a
5. We solve the resulting system of equations by adding them up:
F t2 – F t1 = (m 2 + m 1)
Let's find the acceleration of the bodies: 
- 2 m/s 2
The minus sign means that the real movement occurs with negative acceleration, i.e. the direction of movement is opposite to the chosen direction at the beginning of solving the problem.
Let's find the tension force of the thread: 
= 2.4 N
Task
A mass of 26 kg lies on an inclined plane 13 m long and 5 m high. The friction coefficient is 0.5. What force must be applied to the load along the inclined plane in order to:
a) pull the load evenly;
b) pull the load evenly.
a) b)
Let us arrange the forces acting on the load. The load is acted upon by a gravity force directed vertically downwards, an elastic force directed perpendicular to the interacting surfaces and, when the load moves along an inclined plane, a sliding friction force directed opposite to the speed of the body. In addition, an external force is also applied to the body, which carries out uniform movement of the body along the inclined plane.
For uniform motion necessary (this follows from Newton's 1st law) next condition: the sum of all forces acting on the body is zero. 
F= 218.8 N
- We use the same procedure (Fig. 57b).
In this case, the sliding friction force is directed upward, i.e. in the direction opposite to the speed of the body. Let us write down the condition for the uniform movement of a load down an inclined plane:
In projections onto the x axis:
F + F strand x - F Tr = 0
Rotational movement around a fixed axis is another special case movement solid.
Rotational movement of a rigid body around a fixed axis
it is called such a movement in which all points of the body describe circles, the centers of which are on the same straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular rotation axis
(Fig.2.4).
In technology, this type of motion occurs very often: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity
. Each point of a body rotating around an axis passing through the point ABOUT, moves in a circle, and different points travel different paths over time. So, , therefore the modulus of the point velocity A more than a point IN (Fig.2.5). But the radii of the circles rotate through the same angle over time. Angle - the angle between the axis OH and radius vector, which determines the position of point A (see Fig. 2.5).
Let the body rotate uniformly, i.e., rotate through equal angles at any equal intervals of time. The speed of rotation of a body depends on the angle of rotation of the radius vector, which determines the position of one of the points of the rigid body for a given period of time; it is characterized angular velocity
.
For example, if one body rotates through an angle every second, and the other through an angle, then we say that the first body rotates 2 times faster than the second.
Angular velocity of a body during uniform rotation
is a quantity equal to the ratio of the angle of rotation of the body to the period of time during which this rotation occurred.
We will denote the angular velocity by the Greek letter ω
(omega). Then by definition
Angular velocity is expressed in radians per second (rad/s).
For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of a grinding disk is about 140 rad/s 1 .
Angular velocity can be expressed through rotation speed
, i.e. the number of full revolutions in 1s. If a body makes (Greek letter “nu”) revolutions in 1s, then the time of one revolution is equal to seconds. This time is called rotation period
and denoted by the letter T. Thus, the relationship between frequency and rotation period can be represented as:
A complete rotation of the body corresponds to an angle. Therefore, according to formula (2.1)
If, with uniform rotation, the angular velocity is known in starting moment time rotation angle, then the angle of rotation of the body during time t according to equation (2.1) is equal to:
If , then , or 
.
Angular velocity takes positive values if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the axis OH increases, and negative when it decreases.
Thus, we can describe the position of the points of a rotating body at any time.
Relationship between linear and angular velocities.
The speed of a point moving in a circle is often called linear speed
, to emphasize its difference from angular velocity.
We have already noted that when a rigid body rotates, its different points have unequal linear velocities, but the angular velocity is the same for all points.
There is a relationship between the linear speed of any point of a rotating body and its angular speed. Let's install it. A point lying on a circle of radius R, per revolution will go the way. Since the time of one revolution of a body is a period T, then the modulus of the linear velocity of the point can be found as follows:
1st semester.
1. Material point (particle) - the simplest physical model in mechanics - a body with mass, size, shape, rotation and internal structure which can be neglected under the conditions of the problem under study. The position of a material point in space is determined as the position of a geometric point .
Coordinate system - a set of definitions that implements coordinate method, that is, a way to determine the position of a point or body using numbers or other symbols. The set of numbers that determine the position of a specific point is called the coordinates of this point .
Frame of reference - this is a combination of a reference body, an associated coordinate system and a time reference system, in relation to which the movement of any bodies is considered.
Path is the distance the body has traveled. Path - scalar quantity. For full description movement, it is necessary to know not only the distance traveled, but also the direction of movement.
Moving is a directed line segment that combines starting position body with its subsequent position. Movement, like path, is denoted by the letter S and measured in meters. But these are two different sizes that need to be distinguished.
Relative motion - this is the movement of a material point/body relative to a moving reference system. In this FR, the radius vector of the body is , the speed of the body is .
2. Speed - vector physical quantity, characterizing the speed of movement and direction of movement of a material point relative to the selected reference system; by definition, equal to the derivative of the radius vector of a point with respect to time.
Uniform and uneven movements .
uniform This is a movement in which, in any equal intervals of time, a body travels equal distances.
Uneven This is a movement in which a body passes through different segments of a path in equal intervals of time.
Velocity addition theorem The speed of movement of a body relative to a fixed frame of reference is equal to the vector sum of the speed of this body relative to a moving frame of reference and the speed (relative to a fixed frame) of that point of the moving frame of reference in which the body is located at a given moment in time.
3. Acceleration - a physical quantity that determines the rate of change in the speed of a body, that is, the first derivative of speed with respect to time. Acceleration is vector quantity, showing how much the velocity vector of a body changes during its movement per unit time:
Uniformly accelerated motion - movement in which the acceleration is constant in magnitude and direction.
Rectilinear uniformly accelerated motion – the simplest type uneven movement, in which the body moves along a straight line, and its speed changes equally over any equal periods of time.
You can calculate the acceleration of a body moving rectilinearly and uniformly accelerated using an equation that includes projections of the acceleration and velocity vectors:
v x – v 0x
a x = ---
t
4.Curvilinear movement - the movement of a point along a trajectory that is not a straight line, with arbitrary acceleration and arbitrary speed at any time (for example, movement in a circle).
Angle of rotation - this is not a geometric, but a physical quantity that characterizes the rotation of a body or the rotation of a ray emanating from the center of rotation of the body relative to another ray considered stationary. This is a characteristic of the rotational form of movement, only evaluated in units of a plane angle.
Angular and linear speed.
Angular velocity is a physical quantity equal to the ratio of the angle of rotation to the time interval during which this rotation occurred.
Each point on the circle moves at a certain speed . This speed is called linear . The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinding machine move, repeating the direction of instantaneous speed.
5. Normal and tangential acceleration.
1.Centripetal acceleration - component of the acceleration of a point, characterizing the speed of change in the direction of the velocity vector for a trajectory with curvature. Directed towards the center of curvature of the trajectory, which is where the term comes from. The value is equal to the square of the speed divided by the radius of curvature. The term " centripetal acceleration" is equivalent to the term " normal acceleration ».
2.Tangential acceleration - acceleration component directed tangentially to the motion trajectory. Characterizes the change in the velocity module in contrast to the normal component, which characterizes the change in the direction of velocity.
Full acceleration point is composed of tangential and normal accelerations according to the rule of vector addition. It will always be directed towards the concavity of the trajectory, since normal acceleration is also directed in this direction.
Oscillation period - smallest gap the time during which the oscillator makes one complete oscillation (that is, it returns to the same state in which it was at the initial moment, chosen arbitrarily).
Frequency - a physical quantity, a characteristic of a periodic process, equal to the number of repetitions or occurrences of events (processes) per unit of time. It is calculated as the ratio of the number of repetitions or occurrence of events (processes) to the period of time during which they occurred.
6.Weight, physical quantity, one of the main characteristics of matter, determining its inertial and gravitational properties. Accordingly, a distinction is made between inert and gravitational (heavy, gravitating) materials.
Weight - the force of the body acting on a support (or suspension or other type of fastening), preventing a fall, arising in the field of gravity.
Weightlessness - a state in which the force of interaction between the body and the support (body weight), arising in connection with gravitational attraction, the action of other mass forces, in particular the force of inertia that arises when accelerated movement body, missing.
7. Friction force - This is a force that arises when two bodies come into contact and interferes with their relative motion. The cause of friction is the roughness of the rubbing surfaces and the interaction of the molecules of these surfaces. The force of friction depends on the material of the rubbing surfaces and how tightly these surfaces are pressed against each other.
Types of friction.
1. Sliding friction- a force that arises during the translational movement of one of the contacting/interacting bodies relative to another and acts on this body in the direction opposite direction slip.
2. Rolling friction- moment of force that occurs when one of two contacting/interacting bodies rolls relative to the other.
3. Rest friction- force that arises between two contacting bodies and prevents the occurrence relative motion. This force must be overcome in order to set two contacting bodies in motion relative to each other. Occurs during micromovements (for example, during deformation) of contacting bodies. It acts in the direction opposite to the direction of possible relative motion.
Ground reaction force - it is a force or system of forces expressing the mechanical action of support on a structure that rests on these supports .
8. Deformation - change mutual position particles of a body associated with their movement relative to each other. Deformation is the result of changes in interatomic distances and rearrangement of blocks of atoms. Typically, deformation is accompanied by a change in the magnitude of interatomic forces, the measure of which is elastic mechanical stress.
Types of deformation.
1. Tension - compression - in the resistance of materials - a type of longitudinal deformation of a rod or beam that occurs if a load is applied to it along its longitudinal axis (the resultant of the forces acting on it is normal to the cross section of the rod and passes through its center of mass).
2.Shift - in the resistance of materials - a type of longitudinal deformation of a beam that occurs if a force is applied touching its surface (in this case Bottom part the bar is fixed motionless).
3. Bend - in the resistance of materials, a type of deformation in which there is a curvature of the axes of straight beams or a change in the curvature of the axes of curved beams, a change in the curvature/curvature of the middle surface of the plate or shell. Bending is associated with the occurrence in cross sections beam or shell bending moments.
4.Torsion- one of the types of body deformation. Occurs when a load is applied to a body in the form of a pair of forces in its transverse plane. In this case, only one internal force factor appears in the cross sections of the body - torque. Tension-compression springs and shafts work for torsion.
Elastic force - a force that arises in a body as a result of its deformation and tends to return the body to its original state.
Hooke's law - a statement according to which the deformation that occurs in an elastic body (spring, rod, console, beam, etc.) is proportional to the force applied to this body. Discovered in 1660 by the English scientist Robert Hooke. It should be borne in mind that Hooke's law is satisfied only for small deformations. When the proportionality limit is exceeded, the relationship between stress and strain becomes nonlinear. For many media, Hooke's law is not applicable even at small deformations.
For a thin tensile rod, Hooke's law has the form:
9. Newton's first law postulates the existence inertial systems countdown. Therefore it is also known as the Law of Inertia. Inertia is the property of a body to maintain the speed of its movement unchanged (both in magnitude and direction) when no forces act on the body. To change the speed of a body, it must be acted upon with some force. Naturally, the result of the action of forces of equal magnitude on different bodies will be different. Thus, they say that bodies have different inertia. Inertia is the property of bodies to resist changes in their speed. The amount of inertia is characterized by body weight.
10. Pulse - vector physical quantity, which is a measure mechanical movement bodies. IN classical mechanics body impulse equal to the product masses m of this body at its speed v, the direction of the impulse coincides with the direction of the velocity vector:
Law of conservation of momentum States that vector sum of the impulses of all bodies of the system is a constant value if the vector sum external forces, acting on the system of bodies, is equal to zero.
In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws it can be shown that when a system moves in empty space, momentum is conserved in time, and if there is external influence the rate of change of momentum is determined by the sum of the applied forces.
