1. Quite often one can observe a movement of a body in which its trajectory is a circle. For example, a point on the rim of a wheel moves along a circle as it rotates, points on rotating parts of machine tools, the end of a clock hand, a child sitting on some figure of a rotating carousel.
When moving in a circle, not only the direction of the body’s velocity can change, but also its modulus. Movement is possible in which only the direction of velocity changes, and its magnitude remains constant. This movement is called uniform movement of the body in a circle. Let us introduce the characteristics of this movement.
2. The circular motion of a body is repeated at certain intervals equal to the period of revolution.
The period of revolution is the time during which a body completes one full revolution.
The circulation period is designated by the letter T. The unit of circulation period in SI is taken to be second (1 s).
If during the time t the body has committed N full revolutions, then the period of revolution is equal to:
T = .
The rotation frequency is the number of complete rotations of a body in one second.
The frequency of circulation is indicated by the letter n.
n = .
The unit of circulation frequency in SI is taken to be second to the minus first power (1 s– 1).
Frequency and period of revolution are related as follows:
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n = . |
3. Let's consider a quantity characterizing the position of a body on a circle. Let at the initial moment of time the body be at the point A, and in time t it moved to a point B(Fig. 38).
Let's draw a radius vector from the center of the circle to the point A and radius vector from the center of the circle to the point B. When a body moves in a circle, the radius vector will rotate in time t at angle j. Knowing the angle of rotation of the radius vector, you can determine the position of the body on the circle.
Unit of rotation angle of the radius vector in SI - radian (1 rad).
At the same angle of rotation of the radius vector of the point A And B, located at different distances from its center of a uniformly rotating disk (Fig. 39), will travel different paths.
4. When a body moves in a circle, the instantaneous speed is called linear speed.
The linear speed of a body moving uniformly in a circle, while remaining constant in magnitude, changes in direction and at any point is directed tangentially to the trajectory.
The linear velocity module can be determined by the formula:
v = .
Let a body moving in a circle with a radius R, made one full revolution, Then the path it traveled equal to length circles: l= 2p R, and time is equal to the revolution period T. Therefore, the linear speed of the body:
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v = . |
Since T= , then we can write
v= 2p Rn.
The speed of rotation of a body is characterized by angular velocity.
Angular velocity is a physical quantity equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred.
Angular velocity is denoted by w.
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w = . |
The SI unit of angular velocity is radians per second (1 rad/s):
[w] == 1 rad/s.
For a time equal to the circulation period T, the body makes a full revolution and the angle of rotation of the radius vector j = 2p. Therefore, the angular velocity of the body is:
w =or w = 2p n.
Linear and angular velocities are related to each other. Let's write down the ratio of linear speed to angular speed:
== R.
Thus,
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v=w R. |
At the same angular velocity of points A And B, located on a uniformly rotating disk (see Fig. 39), the linear speed of the point A greater than the linear speed of the point B: vA > vB.
5. When a body moves uniformly in a circle, the magnitude of its linear velocity remains constant, but the direction of the velocity changes. Since speed is a vector quantity, a change in the direction of speed means that the body is moving in a circle with acceleration.
Let's find out how this acceleration is directed and what it is equal to.
Let us recall that the acceleration of a body is determined by the formula:
a == ,
where D v- vector of change in body speed.
Acceleration vector direction a coincides with the direction of vector D v.
Let a body moving in a circle with radius R, for a short period of time t moved from point A to the point B(Fig. 40). To find the change in body speed D v, to the point A let's move the vector parallel to itself v and subtract from it v 0, which is equivalent to adding the vector v with vector – v 0 . Vector directed from v 0 k v, and there is a vector D v.
Consider triangles AOB And ACD. Both of them are isosceles ( A.O. = O.B. And A.C. = A.D. because v 0 = v) and have equal angles: _AOB = _CAD(like angles with mutual perpendicular sides: A.O. B v 0 , O.B. B v). Therefore, these triangles are similar and we can write the ratio of the corresponding sides: = .
Since the points A And B located close to each other, then the chord AB is small and can be replaced with an arc. Arc length is the path traveled by a body in time t With constant speed v: AB = vt.
Besides, A.O. = R, DC= D v, AD = v. Hence,
= ;= ;= a.
Where does the acceleration of the body come from?
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a = . |
From Figure 40 it is clear that the smaller the chord AB, the more accurate the direction of vector D v coincides with the radius of the circle. Therefore, the velocity change vector D v and acceleration vector a directed radially towards the center of the circle. Therefore, the acceleration during uniform motion of a body in a circle is called centripetal.
Thus,
When a body moves uniformly in a circle, its acceleration is constant in magnitude and at any point is directed along the radius of the circle towards its center.
Considering that v=w R, we can write another formula for centripetal acceleration:
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a= w 2 R. |
6. Example of problem solution
The rotation frequency of the carousel is 0.05 s–1. A person spinning on a carousel is at a distance of 4 m from the axis of rotation. Determine the man's centripetal acceleration, period of revolution, and angular velocity of the merry-go-round.
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Given: |
Solution |
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n= 0.05 s– 1 R= 4 m |
Centripetal acceleration is equal to: a= w2 R=(2p n)2R=4p2 n 2R. Treatment period: T = . Angular speed of the carousel: w = 2p n. |
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a? T? |
a= 4 (3.14) 2 (0.05s–1) 2 4 m 0.4 m/s 2 ;
T== 20 s;
w = 2 3.14 0.05 s– 1 0.3 rad/s.
Answer: a 0.4 m/s 2 ; T= 20 s; w 0.3 rad/s.
Self-test questions
1. What kind of motion is called uniform circular motion?
2. What is the orbital period called?
3. What is called frequency of circulation? How are period and frequency related?
4. What is linear speed called? How is it directed?
5. What is angular velocity called? What is the unit of angular velocity?
6. How are the angular and linear velocities of a body related?
7. What is the direction of centripetal acceleration? What formula is it calculated by?
Task 9
1. What is the linear speed of a point on the wheel rim if the radius of the wheel is 30 cm and it makes one revolution in 2 s? What is the angular velocity of the wheel?
2. The car speed is 72 km/h. What are the angular speed, frequency and period of revolution of a car wheel if the wheel diameter is 70 cm? How many revolutions will the wheel make in 10 minutes?
3. What is the distance traveled by the end of the minute hand of the alarm clock in 10 minutes, if its length is 2.4 cm?
4. What is the centripetal acceleration of a point on the rim of a car wheel if the diameter of the wheel is 70 cm? The car speed is 54 km/h.
5. A point on the rim of a bicycle wheel makes one revolution in 2 s. The radius of the wheel is 35 cm. What is the centripetal acceleration of the wheel rim point?
1. The movement of a body in a circle is a movement whose trajectory is a circle. For example, the end of a clock hand, the points of a rotating turbine blade, a rotating engine shaft, etc. move in a circle.
When moving in a circle, the direction of speed continuously changes. In this case, the module of the body’s velocity may change or may remain unchanged. Movement in which only the direction of velocity changes, and its magnitude remains constant, is called uniform movement of the body in a circle. Under the body in in this case mean a material point.
2. The movement of a body in a circle is characterized by certain quantities. These include, first of all, the period and frequency of circulation. Period of revolution of a body in a circle\(T\) - the time during which the body makes one full revolution. The period unit is \([\,T\,] \) = 1 s.
Frequency\((n) \) - the number of full rotations of the body in one second: \(n=N/t \) . The unit of circulation frequency is \([\,n\,] \) = 1 s -1 = 1 Hz (hertz). One hertz is the frequency at which a body makes one revolution in one second.
The relationship between frequency and period of revolution is expressed by the formula: \(n=1/T \) .
Let some body moving in a circle move from point A to point B in time \(t\) . The radius connecting the center of the circle with point A is called radius vector. When a body moves from point A to point B, the radius vector will rotate through the angle \(\varphi \) .
The speed of rotation of a body is characterized by corner And linear speed.
Angular velocity \(\omega \) - physical quantity, equal to the ratio the rotation angle \(\varphi \) of the radius vector to the time period during which this rotation occurred: \(\omega=\varphi/t \) . The unit of angular velocity is radian per second, i.e. \([\,\omega\,] \) = 1 rad/s. For a time equal to the period of revolution, the angle of rotation of the radius vector is equal to \(2\pi \) . Therefore \(\omega=2\pi/T \) .
Linear speed of the body\(v\) - the speed with which the body moves along the trajectory. Linear velocity during uniform circular motion is constant in magnitude, varies in direction and is directed tangentially to the trajectory.
Linear speed is equal to the ratio of the path traveled by the body along the trajectory to the time during which this path was traveled: \(\vec(v)=l/t \) . In one revolution, a point travels a path equal to the length of the circle. Therefore \(\vec(v)=2\pi\!R/T \) . The relationship between linear and angular velocity is expressed by the formula: \(v=\omega R \) .
4. The acceleration of a body is equal to the ratio of the change in its speed to the time during which it occurred. When a body moves in a circle, the direction of the speed changes, therefore, the speed difference is not zero, i.e. the body moves with acceleration. It is determined by the formula: \(\vec(a)=\frac(\Delta\vec(v))(t) \)and is directed in the same way as the velocity change vector. This acceleration is called centripetal acceleration.
Centripetal acceleration with uniform motion of a body in a circle - a physical quantity equal to the ratio of the square of the linear speed to the radius of the circle: \(a=\frac(v^2)(R) \) . Since \(v=\omega R \) , then \(a=\omega^2R \) .
When a body moves in a circle, its centripetal acceleration is constant in magnitude and directed towards the center of the circle.
Part 1
1. When a body moves uniformly in a circle
1) only the module of its speed changes
2) only the direction of its speed changes
3) both the module and the direction of its speed change
4) neither the module nor the direction of its speed changes
2. The linear speed of point 1, located at a distance \(R_1 \) from the center of the rotating wheel, is equal to \(v_1 \) . What is the speed \(v_2 \) of point 2 located from the center at a distance \(R_2=4R_1 \) ?
1) \(v_2=v_1 \)
2) \(v_2=2v_1 \)
3) \(v_2=0.25v_1 \)
4) \(v_2=4v_1 \)
3. The period of rotation of a point along a circle can be calculated using the formula:
1) \(T=2\pi\!Rv \)
2) \(T=2\pi\!R/v \)
3) \(T=2\pi v \)
4) \(T=2\pi/v \)
4. The angular speed of rotation of a car wheel is calculated by the formula:
1) \(\omega=a^2R \)
2) \(\omega=vR^2 \)
3) \(\omega=vR\)
4) \(\omega=v/R \)
5. The angular speed of rotation of a bicycle wheel has increased by 2 times. How did the linear speed of the wheel rim points change?
1) increased by 2 times
2) decreased by 2 times
3) increased 4 times
4) has not changed
6. The linear speed of the helicopter rotor blade points decreased by 4 times. How did their centripetal acceleration change?
1) has not changed
2) decreased by 16 times
3) decreased by 4 times
4) decreased by 2 times
7. The radius of motion of the body in a circle was increased by 3 times, without changing its linear speed. How did the centripetal acceleration of the body change?
1) increased 9 times
2) decreased by 9 times
3) decreased by 3 times
4) increased 3 times
8. What is the rotation period of the engine crankshaft if it makes 600,000 revolutions in 3 minutes?
1) 200,000 s
2) 3300 s
3) 3·10 -4 s
4) 5·10 -6 s
9. What is the rotation frequency of the wheel rim point if the rotation period is 0.05 s?
1) 0.05 Hz
2) 2 Hz
3) 20 Hz
4) 200 Hz
10. The linear speed of a point on the rim of a bicycle wheel with a radius of 35 cm is 5 m/s. What is the period of revolution of the wheel?
1) 14 s
2) 7 s
3) 0.07 s
4) 0.44 s
11.
Establish a correspondence between the physical quantities in the left column and the formulas for their calculation in the right column. In the table under the physical number
values in the left column, write down the corresponding number of the formula you selected from the right column.
PHYSICAL QUANTITY
A) linear speed
B) angular velocity
B) frequency of circulation
FORMULA
1) \(1/T \)
2) \(v^2/R \)
3) \(v/R \)
4) \(\omega R \)
5) \(1/n \)
12.
The period of revolution of the wheel has increased. How the angular and linear velocities of a point on the wheel rim and its centripetal acceleration have changed. Establish a correspondence between the physical quantities in the left column and the nature of their change in the right column.
In the table, under the number of the physical quantity in the left column, write down the corresponding number of the element of your choice in the right column.
PHYSICAL QUANTITY
A) angular velocity
B) linear speed
B) centripetal acceleration
NATURE OF CHANGE IN VALUE
1) increased
2) decreased
3) has not changed
Part 2
13. How far will the wheel rim point travel in 10 s if the rotation frequency of the wheel is 8 Hz and the radius of the wheel is 5 m?
Answers
Law. All movements occur equally in reference systems at rest, or moving relative to each other at a constant speed. This is the principle of sameness or equivalence of inertial frames of reference or Galileo's principle of independence.
General laws movement
1 Law. If the body is not acted upon by other bodies, it maintains a state of rest or uniform rectilinear motion. This is the law of inertia, Newton's first law.
3 Law. All movements of the material body occur independently of each other and add up as vector quantities. So, any body on earth simultaneously participates in the movement of the Sun with the planets around the Galaxy Center at a speed of about 200 km/sec, in the movement of the Earth in orbit at a speed of about 30 km/sec, in the rotation of the Earth around its axis at a speed of up to 400 m/sec and possibly in other movements. The result is a very intricate curvilinear trajectory!
If a body is thrown with an initial speed Vo, at an angle a to the horizon, then the flight range –S is calculated by the formula:
S = 2 V*SIN(a) * COS(a) / g = V*SIN(2a) / g
Maximum range at a =45 degrees. The maximum flight altitude –h is calculated by the formula:
h = V* SIN(a)/2g
Both of these formulas can be obtained by taking into account that the vertical component Vo*SIN(a), and horizontal Vo * COS(a), V =g*t, t =V/g.
Let's make a substitution in the basic formula for height
h = g t/2 = g* (V/g)/2 = V/2g = V* SIN(a)/2g.
This is the required formula. The maximum height when thrown vertically upwards, while
a =90 degrees, SIN(a) =1; h = V*/2g
To derive the formula for flight range, you need to multiply the horizontal component by twice the time of falling from a height h. If you take into account air resistance, the path will be shorter. For a projectile, for example, almost twice. The same range will correspond to two different angles throwing.
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Fig. 11 Flight trajectories of a body thrown at an angle to the horizon. The drawing on the right is a movement in a circle.
w- Angular velocity of a rotating body; radian/sec
b - Angular position of the rotating body; radians or degrees relative to an axis. Radian is the angle at which an arc equal to the radius of the circle is visible from the center of the circle, respectively rad = 360/6.28 = 57.32 degrees
a-angular acceleration is measured in rad/sec 2
b = bo + w * t, Angular movement from bo.
S = b *R - Linear movement along a circle of radius R.
w =(b - bo)/(t –to); - Angular velocity . V = w* R – Circumferential speed
T = 2*p/w =2*p*R/V Hence V = 2*p*R/T
a =ao + w/t – Angular acceleration. Angular acceleration is determined by tangential force and in its absence there will be uniform motion of the body in a circle. In this case, the body is affected by centripetal acceleration, which during a revolution changes the speed by 2*p times. Its value is determined by the formula. a =DV/T =2*p*V/2*p*R/V =V/R
Average values of speed and acceleration do not allow one to calculate the position of a body during uneven movement. To do this, it is necessary to know the values of speed and acceleration in short periods of time or instantaneous values. Instantaneous values are determined through derivatives or differentials.
PHYSICAL QUANTITIES CHARACTERIZING THE CIRCULAR MOTION OF A BODY.
1. PERIOD (T) - the period of time during which the body makes one full revolution.
, where t is the time during which N revolutions are completed.
2. FREQUENCY () - the number of revolutions N made by a body per unit of time.
(hertz)
3. RELATIONSHIP OF PERIOD AND FREQUENCY:
4. MOVE () is directed along chords.
5. ANGULAR MOVEMENT (angle of rotation).
UNIFORM CIRCULAR MOTION is a movement in which the speed module does not change.
6. LINEAR SPEED (directed tangentially to the circle.
7. ANGULAR SPEED 
8. RELATIONSHIP OF LINEAR AND ANGULAR SPEED
Angular velocity does not depend on the radius of the circle along which the body moves. If the problem considers the movement of points located on the same disk, but at different distances from its center, then we must keep in mind that the ANGULAR VELOCITY OF THESE POINTS IS THE SAME.
9. CENTRIPTIPAL (normal) ACCELERATION ().
Since when moving in a circle, the direction of the velocity vector constantly changes, the movement in the circle occurs with acceleration. If a body moves uniformly around a circle, then it has only centripetal (normal) acceleration, which is directed radially towards the center of the circle. Acceleration is called normal, since at a given point the acceleration vector is located perpendicular (normal) to the linear velocity vector. .
If a body moves in a circle with a speed varying in magnitude, then along with normal acceleration, characterizing the change in speed in direction, TANGENTIAL ACCELERATION appears, characterizing the change in speed modulo (). The tangential acceleration is directed tangent to the circle. The total acceleration of a body during uneven circular motion is determined by the Pythagorean theorem:
RELATIVITY OF MECHANICAL MOTION
When considering the motion of a body relative to different systems the reference trajectory, path, speed, movement turn out to be different. For example, a person is sitting on a moving bus. Its trajectory relative to the bus is a point, and relative to the Sun - an arc of a circle, path, speed, displacement relative to the bus are equal to zero, and relative to the Earth they are different from zero. If the motion of a body relative to a moving and stationary reference system is considered, then according to the classical law of addition of velocities, the speed of a body relative to a stationary reference system is equal to the vector sum of the speed of the body relative to a moving reference system and the speed of a moving reference system relative to a stationary one:
Likewise
SPECIAL CASES OF USING THE LAW OF SPEED ADDITION
1) Movement of bodies relative to the Earth
b) bodies move towards each other
2) Movement of bodies relative to each other
a) bodies move in one direction
b) bodies move in different directions(towards each other)
3) Body speed relative to the shore when moving
a) downstream
b) against the current, where is the speed of the body relative to the water, and is the speed of the current.
4) The velocities of the bodies are directed at an angle to each other.
For example: a) a body swims across a river, moving perpendicular to the flow
b) the body swims across the river, moving perpendicular to the shore
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c) the body simultaneously participates in translational and rotational motion, for example, the wheel of a moving car. Each point of the body has a translational speed directed in the direction of the body’s movement and a rotational speed directed tangentially to the circle. Moreover, to find the speed of any point relative to the Earth, it is necessary to vectorially add the speed of translational and rotational motion:
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DYNAMICS
NEWTON'S LAWS
NEWTON'S FIRST LAW (LAW OF INERTIA)
There are such reference systems relative to which the body is at rest or moves rectilinearly and uniformly, if other bodies do not act on it or the actions of the bodies are compensated (balanced).
The phenomenon of maintaining the speed of a body in the absence of the action of other bodies on it or when compensating for the action of other bodies is called inertia.
The reference systems in which Newton's laws are satisfied are called inertial reference systems (IRS). ISO refers to reference systems associated with the Earth or not having acceleration relative to the Earth. Frames of reference moving with acceleration relative to the Earth are non-inertial, and Newton's laws are not satisfied in them. According to Galileo’s classical principle of relativity, all IFRs are equal, the laws of mechanics have the same form in all IFRs, all mechanical processes proceed in the same way in all IFRs (no mechanical experiments carried out inside the IFR can determine whether it is at rest or moving rectilinearly and uniformly).
NEWTON'S SECOND LAW
The speed of a body changes when a force is applied to the body. Any body has the property of inertia . Inertia – This is a property of bodies, consisting in the fact that it takes time to change the speed of a body; the speed of a body cannot change instantly. The body that changes its speed more under the action of the same force is less inert. The measure of inertia is body mass.
The acceleration of a body is directly proportional to the force acting on it and inversely proportional to the mass of the body.
Force and acceleration are always co-directional. If several forces act on a body, then the acceleration imparts to the body resultant these forces (), which is equal to the vector sum of all forces acting on the body: 
If a body makes uniformly accelerated motion, then a constant force acts on it.
NEWTON'S THIRD LAW
Forces arise when bodies interact.
Bodies act on each other with forces directed along the same straight line, equal in magnitude and opposite in direction.
Features of forces arising during interaction:
1. Forces always arise in pairs.
2 The forces arising during interaction are of the same nature.
3. Forces do not have a resultant, because they are applied to different bodies.
FORCES IN MECHANICS
UNIVERSAL GRAVITATION is the force with which all bodies in the Universe are attracted.
LAW OF UNIVERSAL GRAVITY: bodies attract each other with forces directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
(the formula can be used to calculate the attraction of point bodies and balls), where G is the gravitational constant (universal gravity constant), G = 6.67·10 -11, is the mass of bodies, R is the distance between bodies, measured between the centers of bodies. 
GRAVITY – the force of attraction of bodies towards the planet. Gravity is calculated using the formulas:
1) , where is the mass of the planet, is the mass of the body, is the distance between the center of the planet and the body.
2) , where is the acceleration of free fall,
The force of gravity is always directed towards the center of gravity of the planet.
The radius of the orbit of an artificial satellite, - the radius of the planet, - the height of the satellite above surface of the planet,
A body becomes an artificial satellite if it is given the required speed in the horizontal direction. The speed required for a body to move in a circular orbit around a planet is called first escape velocity. To obtain a formula for calculating the first cosmic velocity, it is necessary to remember that all cosmic bodies, including artificial satellites, move under the influence of universal gravity, in addition, velocity is a kinematic quantity; the formula following from Newton's second law Equating the right-hand sides of the formulas, we obtain: or Considering that the body moves in a circle and therefore has centripetal acceleration, we obtain: or. From here - formula for calculating the first escape velocity. Considering that the formula for calculating the first escape velocity can be written as: .Similarly, using Newton’s second law and formulas curvilinear movement, you can determine, for example, the period of revolution of a body in orbit.
ELASTIC FORCE is a force acting on the part of a deformed body and directed in the direction opposite to the displacement of particles during deformation. The elastic force can be calculated using Hooke's law: elastic force is directly proportional to elongation: where is the elongation,
Hardness, . Rigidity depends on the material of the body, its shape and size.
SPRING CONNECTION
Hooke's law is valid only for elastic deformations of bodies. Elastic deformations are those in which, after the cessation of the force, the body acquires its previous shape and size.