Let a material point move uniformly around a circle. Then the modulus of its velocity does not change ($v=const$). But this does not mean that the acceleration of a material point is zero. The velocity vector is directed tangentially to the trajectory of the point. When moving around a circle, the speed changes its direction constantly. This means that the point is moving with acceleration.
Let's consider points A and B belonging to the trajectory of the body in question. The velocity change vector for these points is equal to:
\[\Delta \overline(v)=(\overline(v))"-\overline(v)\left(1\right).\]
If the time of movement between points A and B is short, then the arc AB differs little from the chord AB. Triangles AOB and BMN are similar, therefore:
\[\frac(\Delta v)(v)=\frac(\Delta l)(r)=\alpha \left(2\right).\]
We find the average acceleration module as:
\[\left\langle a\right\rangle =\frac(\Delta v)(\Delta t)=\frac(v\Delta l)(r\Delta t)\left(3\right).\]
The magnitude of the instantaneous acceleration can be obtained by passing to the limit at $\Delta t\to 0\ $ from $\left\langle a\right\rangle $:
The average acceleration vector makes an angle equal to the velocity vector:
\[\beta =\frac(\pi +\alpha )(2)\left(5\right).\]
At $\Delta t\to 0\ $ angle $\alpha \to 0.$ It turns out that the instantaneous acceleration vector makes an angle $\frac(\pi )(2)$ with the velocity vector.
We found that a material point moving uniformly around a circle has an acceleration directed towards the center of the motion trajectory (perpendicular to the velocity vector), its magnitude is equal to the squared speed divided by the radius of the circle. This acceleration is called centripetal or normal, it is usually denoted by $(\overline(a))_n$.
where $\omega $ is the angular velocity of motion of a material point ($v=\omega \cdot r$).
Definition of centripetal acceleration
Definition
So, centripetal acceleration(in the general case) is a component of the total acceleration of a material point, which characterizes how quickly the direction of the velocity vector changes during curvilinear movement. Another component of total acceleration is tangential acceleration, which is responsible for the change in velocity.
Centripetal acceleration is equal to:
\[(\overline(a))_n=\frac(v^2)(r^2)\overline(r\ )\left(7\right),\]
where $e_r=\frac(\overline(r\ ))(r)$ is the unit vector directed from the center of curvature of the trajectory to the point under consideration.
For the first time, the correct formulas for centripetal acceleration were obtained by H. Huygens.
The International System of Units unit of centripetal acceleration is the meter divided by the squared second:
\[\left=\frac(m)(s^2).\]
Examples of problems with solutions
Example 1
Exercise. The disk rotates around a fixed axis. The law of changing the angle of rotation of the radius of the disk sets the equation: $\varphi =5t^2+7\ (rad)$. What is the centripetal acceleration of point A of the disk, which is located at a distance of $r=$0.5 m from the axis of rotation at the end of the fourth second from the start of rotation?
Solution. Let's make a drawing.
The modulus of centripetal acceleration is equal to: \
We find the angular velocity of rotation of the point as:
\[\omega =\frac(d\varphi )(dt)\ (1.2)\]
equation for changing the angle of rotation depending on time:
\[\omega =\frac(d\left(5t^2+7\right))(dt)=10t\ \left(1.3\right).\]
At the end of the fourth second, the angular velocity is:
\[\omega \left(t=4\right)=10\cdot 4=40\ \left(\frac(rad)(s)\right).\]
Using expression (1.1) we find the value of centripetal acceleration:
Answer.$a_n=800\frac(m)(s^2)$.
Example 2
Exercise. The motion of a material point is specified using the equation: $\overline(r)\left(t\right)=0.5\ (\overline(i)(\cos \left(\omega t\right)+\overline(j) (\sin (\omega t)\ )\ ))$, where $\omega =2\ \frac(rad)(s)$. What is the magnitude of the normal acceleration of a point?
Solution. As a basis for solving the problem, we will take the definition of centripetal acceleration in the form:
From the conditions of the problem it is clear that the trajectory of the point is a circle. In parametric form, the equation is: $\overline(r)\left(t\right)=0.5\ (\overline(i)(\cos \left(\omega t\right)+\overline(j)(\sin (\omega t)\ )\ ))$, where $\omega =2\ \frac(rad)(s)$ can be represented as:
\[\left\( \begin(array)(c) x=0.5(\cos \left(2t\right);;\ ) \\ y=0.5(\sin \left(2t\right) .\ ) \end(array) \right.\]
The trajectory radius can be found as:
The velocity components are equal:
\ \
Let's get the speed module:
Substitute the speed value and the radius of the circle into expression (2.2), we have:
Answer.$a_n=2\frac(m)(s^2)$.
Two rays emanating from it form an angle. Its value can be defined in both radians and degrees. Now, at some distance from the center point, let’s mentally draw a circle. The measure of angle, expressed in radians, is then the mathematical ratio of the length of the arc L, separated by two rays, to the value of the distance between the center point and the line of the circle (R), that is:
If we now imagine the described system as material, then we can apply to it not only the concept of angle and radius, but also centripetal acceleration, rotation, etc. Most of them describe the behavior of a point located on a rotating circle. By the way, a solid disk can also be represented by a set of circles, the difference of which is only in the distance from the center.
One of the characteristics of such a rotating system is its orbital period. It indicates the time value during which a point on an arbitrary circle will return to its initial position or, which is also true, turn 360 degrees. At a constant rotation speed, the correspondence T = (2*3.1416) / Ug is satisfied (hereinafter Ug is the angle).
Rotation speed indicates the number of complete revolutions performed in 1 second. At a constant speed we get v = 1 / T.
Depends on time and the so-called rotation angle. That is, if we take an arbitrary point A on the circle as the origin, then when the system rotates, this point will move to A1 in time t, forming an angle between the radii A-center and A1-center. Knowing the time and angle, you can calculate the angular velocity.
And since there is a circle, movement and speed, it means that centripetal acceleration is also present. It represents one of the components that describe movement in the case of curvilinear movement. The terms "normal" and "centripetal acceleration" are identical. The difference is that the second is used to describe movement in a circle when the acceleration vector is directed towards the center of the system. Therefore, it is always necessary to know exactly how the body (point) moves and its centripetal acceleration. Its definition is as follows: it is the rate of change of speed, the vector of which is directed perpendicular to the direction of the vector and changes the direction of the latter. The encyclopedia states that Huygens studied this issue. The formula for centripetal acceleration proposed by him looks like:
Acs = (v*v) / r,
where r is the radius of curvature of the traveled path; v - moving speed.
The formula used to calculate centripetal acceleration still causes heated debate among enthusiasts. For example, an interesting theory was recently voiced.
Huygens, considering the system, proceeded from the fact that the body moves in a circle of radius R with a speed v measured at the initial point A. Since the inertia vector is directed along, a trajectory is obtained in the form of a straight line AB. However, the centripetal force holds the body on the circle at point C. If we mark the center as O and draw lines AB, BO (the sum of BS and CO), as well as AO, we get a triangle. According to the Pythagorean law:
BS=(a*(t*t)) / 2, where a is acceleration; t - time (a*t*t is the speed).
If we now use the Pythagorean formula, then:
R2+t2+v2 = R2+(a*t2*2*R) / 2+ (a*t2/2)2, where R is the radius, and the alphanumeric spelling without the multiplication sign is the degree.
Huygens admitted that since the time t is small, it can be ignored in the calculations. Having transformed the previous formula, she came to the well-known Acs = (v*v) / r.
However, since time is taken squared, a progression arises: the larger t, the higher the error. For example, for 0.9 almost the total value of 20% is unaccounted for.
The concept of centripetal acceleration is important for modern science, but, obviously, it is too early to put an end to this issue.
Since linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.
Angular velocity
Let's choose a point on the circle 1 . Let's build a radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation frequency is the number of revolutions per second.
Frequency and period are interrelated by the relationship
Relationship with angular velocity
Linear speed
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.
Consider a point on a circle that makes one revolution, the time spent is the period T. The path that a point travels is the circumference.
Centripetal acceleration
When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.
Using the previous formulas, we can derive the following relationships
Points lying on the same straight line emanating from the center of the circle (for example, these could be points that lie on the spokes of a wheel) will have the same angular velocities, period and frequency. That is, they will rotate the same way, but with different linear speeds. The further a point is from the center, the faster it will move.
The law of addition of speeds is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.
The Earth participates in two main rotational movements: diurnal (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, the cause of any acceleration is force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then the acting force is the elastic force.
If a body lying on a disk rotates with the disk around its axis, then such a force is the friction force. If the force stops its action, then the body will continue to move in a straight line
Consider the movement of a point on a circle from A to B. The linear speed is equal to vA And vB respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.
Problem on applying the equation of state of an ideal gas
Ticket 4
Movement in a circle with a constant absolute speed; period and frequency; centripetal acceleration.
When a body moves uniformly in a circle, the velocity module remains constant, and the direction of the velocity vector changes during the movement. The motion of a body in a circle can be described by specifying the angle of rotation of the radius. The rotation angle is measured in radians. The ratio of the angle of rotation of radius φ to the period of time during which this rotation is made is called angular velocity: ω = φ / t . Linear speed is the ratio of the length of the traveled path l to the time interval t:v = l/t. There is the following relationship between linear and angular speed:v =ω · R. When a body moves in a circle, the direction of speed changes, therefore, the body moves with acceleration, which is called centripetal:a =v 2 /R. Circular motion is characterized by period and frequency. Period is the time of one revolution. Frequency is the number of revolutions per second. There is a relationship between period and frequency:T = 1/v . The frequency and period can be found through the angular velocity: ω =2 π υ = 2 π / T.
2. Electric current in solutions and melts of electrolytes: Faraday’s law; determination of the charge of a monovalent ion; technical applications of electrolysis.
Electrolytes– aqueous solutions of salts, acids and alkalis. Electrolytic dissociation- the process of decomposition of electrolyte molecules into ions during the dissolution of electrolytes under the influence of the electric field of polar water molecules. Degree of dissociation, i.e. the proportion of molecules in a solute that break up into ions depends on temperature, solution concentration, and dielectric constant of the solvent. With increasing temperature, the degree of dissociation increases and, consequently, the concentration of positively and negatively charged ions increases. When ions of different signs meet, they can again unite into neutral molecules - recombine. Charge carriers in aqueous solutions or melts of electrolytes are positively or negatively charged ions. Since charge transfer in aqueous solutions or electrolyte melts is carried out by ions, such conductivity is called ionic. Electric current in solutions and melts of electrolytes- this is the ordered movement of positive ions to the cathode, and negative ions to the anode.
Electrolysis is the process of release of a pure substance at the electrode associated with redox reactions.
Faraday formulated the law of electrolysis: m = q · t.
The mass of the substance released from the electrolyte on the electrodes turns out to be greater, the greater the charge passed through the electrolyte q, or I · t, where I is the current strength, t is the time of its passage through the electrolyte. The coefficient k, which turns this proportionality into the equality m =k · I · t, is called the electrochemical equivalent of the substance.
Electrolysis is used:
1. Galvanoplasty, i.e. copying relief objects.
2. Galvanostegy, i.e. applying a thin layer of another metal (chrome, nickel, gold) to metal products.
3. Purification of metals from impurities (metal refining).
4. Electropolishing of metal products. In this case, the product plays the role of an anode in a specially selected electrolyte. On microroughnesses (protrusions) on the surface of the product, the electrical potential increases, which contributes to their priority dissolution in the electrolyte.
5. Obtaining some gases (hydrogen, chlorine).
6. Obtaining metals from ore melts. This is how aluminum is mined.
Problem on applying gas laws.
Ticket 5
1. Newton's first law: inertial frame of reference.
Newton's first law:there are frames of reference relative to which a body retains its speed unchanged if other bodies do not act on it or the actions of other bodies compensate each other. Such reference systems are called inertial. Thus, all bodies that are not acted upon by other bodies move each other. relative to a friend evenly and straight and the frame of reference associated with any of them, is inertial. Newton's first law is sometimes called the law of inertia(inertia - the phenomenon that the speed of a body remains unchanged at absence of external influences on the body or their compensation).
2. Electric current in semiconductors: dependence of the resistance of semiconductors on external conditions; intrinsic conductivity of semiconductors; donor and acceptor impurities; r-p-transition; semiconductor diodes.
Semiconductors include substances whose resistivity is intermediate between conductors and dielectrics. Conductivity of pure semiconductors in the absence of impurities called intrinsic conductivity , since it is determined by the properties of the semiconductor itself. There are two mechanisms of intrinsic conductivity - electronic and hole. Electronic conductivity is carried out by the directed movement in the interatomic space of free electrons that have left the valence shell of the atom as a result of heating the semiconductor or under the influence of external fields. It's called a hole a vacant electronic state in an atom, formed when a free electron appears, has a positive charge. The valence electron of a neighboring atom, attracted to a hole, can jump into it (recombine). In this case, a new hole is formed in its original place, which can then similarly move around the crystal.
Hole conductivity is carried out by the directed movement of valence electrons between the electron shells of neighboring atoms to vacant places (holes).
The intrinsic conductivity of semiconductors is usually low, since the number of free charges is small.
Impurities in a semiconductor - atoms of foreign chemical elements contained in the main semiconductor. Dosed introduction of impurities into a pure semiconductor makes it possible to purposefully change its conductivity. Impurity conductivity - conductivity of semiconductors, due to the introduction of impurities into their crystal lattice. By changing the concentration of impurity atoms, you can significantly change the number of charge carriers of one or another sign. The sign of the charge carriers is determined by the valence of the impurity atoms. There are donor and acceptor impurities . The valence of the donor impurity atoms is greater than the valence of the main semiconductor (for example, arsenic). The valence of the acceptor impurity atoms is less than the valence of the main semiconductor (for example, indium). A semiconductor with a donor impurity is called an n-type semiconductor , since it has predominantly electronic conductivity.
A semiconductor with an acceptor impurity is called a p-type semiconductor , since the hole has a positive charge. A special layer is formed at the point of contact of impurity semiconductors R- n - transition -contact layer of two impurity semiconductors p- and n-type. A characteristic feature of a pn junction is its one-way conductivity: it passes current almost only in one direction. The field strength of this blocking layer is directed from the n- to the p-semiconductor (from plus to minus), preventing further separation of charges. Barrier layer- a double layer of opposite electric charges that creates an electric field at the transition, preventing the free separation of charges.
Semiconductor diode - an element of an electrical system containing a pn junction and two terminals for inclusion in an electrical circuit.
The ability of a pn junction to pass current almost only in one direction is used to convert (with the help of a diode) an alternating current that changes its direction into a direct (more precisely pulsating) current in one direction.
Transistor - a semiconductor device with two pn junctions and three terminals for inclusion in an electrical circuit. Serves to convert or amplify alternating current into electricity. schemes.
The transistor forms three thin layers of dopant semiconductors: emitter, base and collector. The emitter is a source of free electrons and is made of a n-type semiconductor. The base regulates the current in the transistor; it is a thin layer (about 10 microns thick) of a p-type semiconductor. The collector, which intercepts the flow of charge carriers from the emitter through the base, is made of a n-type semiconductor. The transistor is used in transistor generators to produce high-frequency electrical oscillations. Semiconductors are small in size, so they are widely used in integrated circuits, being their integral part. Computers, radio, television, space communications, and automation systems are created on the basis of these circuits and can contain up to a million diodes and transistors.
3. Experimental task: “Measuring air humidity using a psychrometer.”
Ticket 6
1. Newton’s second law: the concept of mass and force, the principle of superposition of forces; formulation of Newton's second law; classical principle of relativity.
The interactions differ from each other both quantitatively and qualitatively. For example, it is clear that the more a spring is deformed, the greater the interaction of its coils. Or the closer two like charges are, the stronger they will attract. In the simplest cases of interaction, the quantitative characteristic is force. Force is the reason for the acceleration of bodies (in an inertial frame of reference). Force is a vector physical quantity, which is a measure of the acceleration acquired by bodies during interaction. The resultant of several forces is a force whose action is equivalent to the action of the forces that it replaces. The resultant is the vector sum of all forces applied to the body.
Newton's second law: the vector sum of all forces acting on a body is equal to the product of the mass of the body and the acceleration imparted to this body: F= m a
A force of 1 newton imparts an acceleration of 1 m/s 2 to a body weighing 1 kg.
Thus, all bodies have the property inertia, consisting in the fact that the speed of a body cannot be changed instantly. The measure of a body's inertia is its weight: The greater the mass of the body, the greater the force must be applied to impart the same acceleration to it.
2. Magnetic field: concept of a magnetic field; magnetic induction; magnetic induction lines, magnetic flux; movement of charged particles in a uniform magnetic field.
Interactions between conductors with current, i.e. interactions between moving electric charges, are called magnetic. The forces with which current-carrying conductors act on each other are called magnetic forces.
The magnetic field is a special form of matter through which interaction occurs between moving electrically charged particles.
Properties of magnetic field:
1. The magnetic field is generated by electric current (moving charges).
2. A magnetic field is detected by its effect on electric current (moving charges).
Like the electric field, the magnetic field really exists, regardless of us, of our knowledge about it.
Magnetic induction IN- the ability of a magnetic field to exert a force on a current-carrying conductor (vector quantity). Measured in T (Tesla).
The direction of the magnetic induction vector is taken to be :
- the direction from the south pole S to the north N of a magnetic needle freely positioned in a magnetic field. This direction coincides with the direction of the positive normal to the closed loop with current.
- the direction of the magnetic induction vector is set using gimlet rules:
if the direction of translational movement of the gimlet coincides with the direction of the current in the conductor, then the direction of rotation of the gimlet handle coincides with the direction of the magnetic induction vector.
Magnetic induction lines - graphical representation of a magnetic field.
A line at any point of which the magnetic induction vector is directed along a tangent - the magnetic induction line. A uniform field is parallel lines, a non-uniform field is curved lines. The more lines, the greater the strength of this field. Fields with closed lines of force are called vortex fields. The magnetic field is a vortex field.
Magnetic flux – a value equal to the product of the magnitude of the magnetic induction vector by the area and the cosine of the angle between the vector and the normal to the surface.
Ampere power – the force acting on a conductor in a magnetic field is equal to the product of the magnetic induction vector by the current strength, the length of the conductor section and the sine of the angle between the magnetic induction and the conductor section.
where l is the length of the conductor, B is the magnetic induction vector, I is the current strength.
Ampere force is used in loudspeakers and speakers.
Operating principle: An alternating electric current flows through the coil with a frequency equal to the audio frequency from a microphone or from the output of a radio receiver. Under the action of the Ampere force, the coil oscillates along the axis of the loudspeaker in time with the current fluctuations. These vibrations are transmitted to the diaphragm, and the surface of the diaphragm emits sound waves.
Lorentz force - force acting on a moving charged particle from a magnetic field.
Lorentz force. Since current represents the ordered movement of electric charges, it is natural to assume that the Ampere force is the resultant of the forces acting on individual charges moving in a conductor. It has been experimentally established that a force actually acts on a charge moving in a magnetic field. This force is called the Lorentz force. The module F l of force is found by the formula
where B is the modulus of induction of the magnetic field in which the charge moves, q and v are the absolute magnitude of the charge and its speed, and a is the angle between the vectors v and B.
This force is perpendicular to the vectors v and B, its direction is along left hand rule : if the hand is positioned so that the four extended fingers coincide with the direction of movement of the positive charge, the magnetic field induction lines enter the palm, then the thumb set 900 away shows the direction of the force. In the case of a negative particle, the direction of the force is opposite.
Since the Lorentz force is perpendicular to the velocity of the particle, it does no work.
The Lorentz force is used in televisions and mass spectrographs.
Operating principle: The vacuum chamber of the device is placed in a magnetic field. Charged particles (electrons or ions) accelerated by an electric field, having described an arc, fall on a photographic plate, where they leave a trace that makes it possible to measure the radius of the trajectory with great accuracy. This radius determines the specific charge of the ion. Knowing the charge of an ion, it is easy to determine its mass.
3. Experimental task: “Constructing a graph of temperature versus water cooling time.”
Ticket 7
1. Newton's third law: formulation; characteristics of action and reaction forces: module, direction, point of application, nature.
Newton's third law:bodies interact with each other with forces directed along one straight line, equal in magnitude and opposite in magnitude
direction:F 12 = - F 21.
The forces included in Newton's third law have same physical nature And do not compensate for each other because applied to different bodies. Thus, forces always exist in pairs: for example, the force of gravity acting on a person from the Earth is related, according to Newton’s III law, to the force with which a person attracts the Earth. These forces are equal in magnitude, but the acceleration of the Earth is many times less than the acceleration of a person, since its mass is much greater.
2.Faraday's law of electromagnetic induction; Lenz's rule; self-induction phenomenon; inductance; magnetic field energy.
Faraday in 1831 established that the emf. induction does not depend on the method of changing the magnetic flux and is determined only by the speed of its change, i.e.
Law of Electromagnetic Induction : The induced emf in a conductor is equal to the rate of change of the magnetic flux passing through the area covered by the conductor. The minus sign in the formula is a mathematical expression of Lenz's rule.
It is known that magnetic flux is an algebraic quantity. Let us assume that the magnetic flux penetrating the area of the circuit is positive. As this flux increases, an emf occurs. induction, under the influence of which an induced current appears, creating its own magnetic field directed towards the external field, i.e. the magnetic flux of the induction current is negative. If the flow penetrating the contour area decreases, then, i.e. the direction of the magnetic field of the induction current coincides with the direction of the external field.
Let's consider one of the experiments carried out by Faraday to detect the induced current, and therefore the emf. induction. If a magnet is pushed or pulled into a solenoid connected to a very sensitive electrical measuring device (galvanometer), then as the magnet moves, a deflection of the galvanometer needle is observed, indicating the occurrence of an induced current. The same thing is observed when the solenoid moves relative to the magnet. If the magnet and the solenoid are stationary relative to each other, then no induced current occurs. From the above experience it follows conclusion, that with the mutual movement of these bodies, a change in the magnetic flux occurs through the turns of the solenoid, which leads to the appearance of an induced current caused by the emerging emf. induction.
The direction of the induction current is determined by Lenz's rule : induced current always has a direction such that the magnetic field it creates prevents the change in magnetic flux that this current causes.
From this rule it follows that as the magnetic flux increases, the resulting induced current has a direction such that the magnetic field generated by it is directed against the external field, counteracting the increase in magnetic flux. A decrease in magnetic flux, on the contrary, leads to the appearance of an induction current, which creates a magnetic field coinciding in direction with the external field.
Application of electromagnetic induction in technology, in industry, for generating electricity at power plants, heating and melting of conductive materials (metals) in induction electric furnaces, etc.
3. Experimental task: “Study of the dependence of the period and frequency of free oscillations of a mathematical pendulum on the length of the thread.”
Ticket 8
1. Body impulse. Law of conservation of momentum: body momentum and force impulse; expression of Newton's second law using the concepts of changes in body momentum and force impulse; law of conservation of momentum; jet propulsion.
The momentum of a body is called a vector physical quantity, which is a quantitative characteristic of the translational motion of bodies. The impulse is designated p. The momentum of a body is equal to the product of the mass of the body and its speed: p = m v. The direction of the momentum vector p coincides with the direction of the body velocity vector v. The impulse unit is kg m/s.
For the momentum of a system of bodies, the conservation law is satisfied, which is valid only for closed physical systems. In general, a closed system is a system that does not exchange energy and mass with bodies and fields that are not part of it. In mechanics, a closed system is a system on which no external forces act or the action of these forces is compensated. In this case, p1 = p2, where p1 is the initial impulse of the system, and p2 is the final one. In the case of two bodies included in the system, this expression has the formm 1 v 1 + m 2 v 2 = m 1 v 1 ´ + m 2 v 2 ´ ,
where m1 and m2 are the masses of bodies, and v1 and v2 are the velocities before the interaction, v1´ and v2´ are the velocities after the interaction. This formula is the mathematical expressionlaw of conservation of momentum:
The momentum of a closed physical system is conserved during any interactions occurring within this system.
In mechanics, the law of conservation of momentum and Newton's laws are interconnected. If a force acts on a body of mass m during time t and the speed of its movement changes from v0 to v, then the acceleration of motion a of the body is equal. Based on Newton’s second law for force F, we can write , it follows
, where Ft is a vector physical quantity that characterizes the action of a force on a body over a certain period of time and is equal to the product of the force and the time of its action, called the impulse of the force. The SI unit of force impulse is N*s.
The law of conservation of momentum underlies jet propulsion.
Jet propulsion - this is the movement of the body that occurs after the separation of its part from the body.
Let a body of mass m be at rest. Some part of it with mass m1 separated from the body at a speed v1. Then the remaining part will move in the opposite direction with a speed ν2, the mass of the remaining part is m2. Indeed, the sum of the impulses of both parts of the body before separation was equal to zero and after separation will be equal to zero:
Much credit for the development of jet propulsion belongs to K.E. Tsiolkovsky
2. Oscillatory circuit. Free electromagnetic oscillations: damping of free oscillations; period of electromagnetic oscillations.
Electromagnetic oscillations are a periodic change in charge, current or voltage.
These changes occur according to the harmonic law:
For charge q =q m ·cos ω 0 ·t; for current i = i m ·cos ω 0 ·t; for voltage u =u m cos ω 0 t, where
q - charge change, C (Coulomb), u - voltage change, V (Volt), i - current change, A (Ampere), q m - charge amplitude, i m - current amplitude; u m - voltage amplitude; ω 0 - cyclic frequency, rad/s; t – time.
Physical quantities characterizing vibrations:
1. Period is the time of one complete oscillation. T, s
2. Frequency - the number of oscillations completed in 1 second, Hz
3. Cyclic frequency - the number of oscillations completed in 2 π seconds, rad/s.
Electromagnetic oscillations can be free or forced:
Free email magnetic oscillations arise in an oscillatory circuit and are damped. Forced emails magnetic oscillations are created by a generator.
If e.l.m. oscillations arise in a circuit of an inductor and a capacitor, then the alternating magnetic field turns out to be connected to the coil, and the alternating electric field is concentrated in the space between the capacitor plates. An oscillatory circuit is a closed connection between a coil and a capacitor. Oscillations in the circuit proceed according to the harmonic law, and the period of oscillations is determined by the Thomson formula.T = 2·π·
Increasing the e.l.m. period fluctuations with increasing inductance and capacitance is explained by the fact that as inductance increases, the current increases more slowly over time and falls more slowly to zero. And the larger the capacity, the longer it takes to recharge the capacitor.
3. Experimental task: “Determination of the refractive index of plastic.”
Allows us to exist on this planet. How can we understand what centripetal acceleration is? The definition of this physical quantity is presented below.
Observations
The simplest example of the acceleration of a body moving in a circle can be observed by rotating a stone on a rope. You pull the rope, and the rope pulls the stone towards the center. At each moment of time, the rope imparts a certain amount of movement to the stone, and each time in a new direction. You can imagine the movement of the rope as a series of weak jerks. A jerk - and the rope changes its direction, another jerk - another change, and so on in a circle. If you suddenly release the rope, the jerking will stop, and with it the change in direction of speed will stop. The stone will move in the direction tangent to the circle. The question arises: “With what acceleration will the body move at this instant?”
Formula for centripetal acceleration
First of all, it is worth noting that the movement of a body in a circle is complex. The stone participates in two types of motion simultaneously: under the influence of force it moves towards the center of rotation, and at the same time along a tangent to the circle, moving away from this center. According to Newton's Second Law, the force holding a stone on a rope is directed toward the center of rotation along the rope. The acceleration vector will also be directed there.
Let us assume that after some time t our stone, moving uniformly with speed V, gets from point A to point B. Let us assume that at the moment of time when the body crossed point B, the centripetal force ceased to act on it. Then, in a period of time, it would get to point K. It lies on the tangent. If at the same moment of time only centripetal forces acted on the body, then during time t, moving with the same acceleration, it would end up at point O, which is located on a straight line representing the diameter of a circle. Both segments are vectors and obey the rule of vector addition. As a result of summing these two movements over a period of time t, we obtain the resulting movement along the arc AB.
If the time interval t is taken to be negligibly small, then the arc AB will differ little from the chord AB. Thus, it is possible to replace movement along an arc with movement along a chord. In this case, the movement of the stone along the chord will obey the laws of rectilinear motion, that is, the distance AB traveled will be equal to the product of the speed of the stone and the time of its movement. AB = V x t.
Let us denote the desired centripetal acceleration by the letter a. Then the path traveled only under the influence of centripetal acceleration can be calculated using the formula for uniformly accelerated motion:
Distance AB is equal to the product of speed and time, that is, AB = V x t,
AO - calculated earlier using the formula of uniformly accelerated motion for moving in a straight line: AO = at 2 / 2.
Substituting this data into the formula and transforming it, we get a simple and elegant formula for centripetal acceleration:
In words, this can be expressed as follows: the centripetal acceleration of a body moving in a circle is equal to the quotient of linear velocity squared by the radius of the circle along which the body rotates. The centripetal force in this case will look like the picture below.
Angular velocity
Angular velocity is equal to the linear velocity divided by the radius of the circle. The converse statement is also true: V = ωR, where ω is the angular velocity
If we substitute this value into the formula, we can obtain an expression for the centrifugal acceleration for the angular velocity. It will look like this:
Acceleration without changing speed
And yet, why does a body with acceleration directed towards the center not move faster and move closer to the center of rotation? The answer lies in the very formulation of acceleration. The facts show that circular motion is real, but to maintain it requires acceleration directed towards the center. Under the influence of the force caused by this acceleration, a change in the amount of motion occurs, as a result of which the trajectory of motion is constantly curved, all the time changing the direction of the velocity vector, but without changing its absolute value. Moving in a circle, our long-suffering stone rushes inward, otherwise it would continue to move tangentially. Every moment of time, going tangentially, the stone is attracted to the center, but does not fall into it. Another example of centripetal acceleration would be a water skier making small circles on the water. The athlete's figure is tilted; he seems to fall, continuing to move and leaning forward.
Thus, we can conclude that acceleration does not increase the speed of the body, since the velocity and acceleration vectors are perpendicular to each other. Added to the velocity vector, acceleration only changes the direction of movement and keeps the body in orbit.
Exceeding the safety factor
In the previous experiment we were dealing with a perfect rope that did not break. But let’s say our rope is the most ordinary, and you can even calculate the force after which it will simply break. In order to calculate this force, it is enough to compare the strength of the rope with the load it experiences during the rotation of the stone. By rotating the stone at a higher speed, you impart to it a greater amount of motion, and therefore greater acceleration.
With a jute rope diameter of about 20 mm, its tensile strength is about 26 kN. It is noteworthy that the length of the rope does not appear anywhere. By rotating a 1 kg load on a rope with a radius of 1 m, we can calculate that the linear speed required to break it is 26 x 10 3 = 1 kg x V 2 / 1 m. Thus, the speed that is dangerous to exceed will be equal to √ 26 x 10 3 = 161 m/s.
Gravity
When considering the experiment, we neglected the effect of gravity, since at such high speeds its influence is negligible. But you can notice that when unwinding a long rope, the body describes a more complex trajectory and gradually approaches the ground.
Celestial bodies
If we transfer the laws of circular motion into space and apply them to the movement of celestial bodies, we can rediscover several long-familiar formulas. For example, the force with which a body is attracted to the Earth is known by the formula:
In our case, the factor g is the same centripetal acceleration that was derived from the previous formula. Only in this case, the role of the stone will be played by a celestial body attracted to the Earth, and the role of the rope will be played by the force of gravity. The g factor will be expressed in terms of the radius of our planet and its rotation speed.
Results
The essence of centripetal acceleration is the hard and thankless work of keeping a moving body in orbit. A paradoxical case is observed when, with constant acceleration, a body does not change the value of its speed. To the untrained mind, such a statement is quite paradoxical. Nevertheless, both when calculating the motion of an electron around the nucleus, and when calculating the speed of rotation of a star around a black hole, centripetal acceleration plays an important role.