Because line speed uniformly changes direction, then the movement along the circle cannot be called uniform, it is uniformly accelerated.
Angular velocity
Pick a point on the circle 1 . Let's build a radius. For a unit of time, the point will move to the point 2 . In this case, the radius describes the angle. The angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T is the time it takes the body to make one revolution.
RPM is the number of revolutions per second.
The frequency and period are related by the relation
Relationship with angular velocity
Line speed
Each point on the circle moves at some 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 grinder moving in the same direction as the instantaneous velocity.
Consider a point on a circle that makes one revolution, the time that is spent - this is the period T.The path that the point overcomes is the circumference of the circle.
centripetal acceleration
When moving along a circle, the acceleration vector is always perpendicular to the velocity vector, directed to the center of the circle.
Using the previous formulas, we can derive the following relations
Points lying on the same straight line emanating from the center of the circle (for example, these can be points that lie on the wheel spoke) will have the same angular velocities, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The farther the point is from the center, the faster it will move.
The law of addition of velocities 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: daily (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 a 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 along with the disk around its axis, then such a force is the force of friction. If the force ceases to act, 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 velocity is equal to
Now let's move on to a fixed system connected to the earth. The total acceleration of point A will remain the same both in absolute value and in direction, since the acceleration does not change when moving from one inertial frame of reference to another. From the point of view of a stationary observer, the trajectory of point A is no longer a circle, but a more complex curve (cycloid), along which the point moves unevenly.
Uniform movement around the circumference is the simplest example. For example, the end of the clock hand moves along the dial along the circle. The speed of a body in a circle is called line speed.
With a uniform motion of the body along a circle, the modulus of the velocity of the body does not change over time, that is, v = const, but only the direction of the velocity vector changes in this case (a r = 0), and the change in the velocity vector in the direction is characterized by a value called centripetal acceleration() a n or a CA. At each point, the centripetal acceleration vector is directed to the center of the circle along the radius.
The module of centripetal acceleration is equal to
a CS \u003d v 2 / R
Where v is the linear speed, R is the radius of the circle
Rice. 1.22. The movement of the body in a circle.
When describing the motion of a body in a circle, use radius turning angle is the angle φ by which the radius drawn from the center of the circle to the point where the moving body is at that moment rotates in time t. The rotation angle is measured in radians. equal to the angle between two radii of the circle, the length of the arc between which is equal to the radius of the circle (Fig. 1.23). That is, if l = R, then
1 radian= l / R
Because circumference is equal to
l = 2πR
360 o \u003d 2πR / R \u003d 2π rad.
Consequently
1 rad. \u003d 57.2958 about \u003d 57 about 18 '
Angular velocity uniform motion of the body in a circle is the value ω, equal to the ratio of the angle of rotation of the radius φ to the time interval during which this rotation is made:
ω = φ / t
The unit of measure for angular velocity is radians per second [rad/s]. The linear velocity modulus is determined by the ratio of the distance traveled l to the time interval t:
v= l / t
Line speed with uniform motion along a circle, it is directed tangentially at a given point on the circle. When the point moves, the length l of the circular arc traversed by the point is related to the angle of rotation φ by the expression
l = Rφ
where R is the radius of the circle.
Then, in the case of uniform motion of the point, the linear and angular velocities are related by the relation:
v = l / t = Rφ / t = Rω or v = Rω
Rice. 1.23. Radian.
Period of circulation- this is the period of time T, during which the body (point) makes one revolution around the circumference. Frequency of circulation- this is the reciprocal of the circulation period - the number of revolutions per unit time (per second). The frequency of circulation is denoted by the letter n.
n=1/T
For one period, the angle of rotation φ of the point is 2π rad, therefore 2π = ωT, whence
T = 2π / ω
That is, the angular velocity is
ω = 2π / T = 2πn
centripetal acceleration can be expressed in terms of the period T and the frequency of revolution n:
a CS = (4π 2 R) / T 2 = 4π 2 Rn 2
When describing the movement of a point along a circle, we will characterize the movement of a point by an angle Δφ , which describes the radius vector of the point in time Δt. Angular displacement in an infinitesimal time interval dt denoted dφ.
Angular displacement is a vector quantity. The direction of the vector (or ) is determined according to the rule of the gimlet: if you rotate the gimlet (screw with right-hand thread) in the direction of the point movement, then the gimlet will move in the direction of the angular displacement vector. On fig. 14 point M moves clockwise, if you look at the plane of movement from below. If you turn the gimlet in this direction, then the vector will be directed upwards.
Thus, the direction of the angular displacement vector is determined by the choice of the positive direction of rotation. The positive direction of rotation is determined by the gimlet rule with right-hand threads. However, with the same success it was possible to take a gimlet with a left-hand thread. In this case, the direction of the angular displacement vector would be opposite.
When considering such quantities as speed, acceleration, displacement vector, the question of choosing their direction did not arise: it was determined in a natural way from the nature of the quantities themselves. Such vectors are called polar. Vectors similar to the angular displacement vector are called axial, or pseudovectors. The direction of the axial vector is determined by the choice of the positive direction of rotation. In addition, the axial vector has no application point. Polar vectors, which we have considered so far, are applied to a moving point. For an axial vector, you can only specify the direction (axis, axis - lat.), along which it is directed. The axis along which the angular displacement vector is directed is perpendicular to the plane of rotation. Typically, the angular displacement vector is drawn on an axis passing through the center of the circle (Fig. 14), although it can be drawn anywhere, including on an axis passing through the point in question.
In the SI system, angles are measured in radians. A radian is an angle whose arc length is equal to the radius of the circle. Thus, the total angle (360 0) is 2π radians.
Moving a point around a circle
Angular velocity is a vector quantity, numerically equal to the angle rotation per unit of time. Usually referred to as the angular velocity Greek letterω. By definition, angular velocity is the derivative of an angle with respect to time:
. (19)
The direction of the angular velocity vector coincides with the direction of the angular displacement vector (Fig. 14). The angular velocity vector, like the angular displacement vector, is an axial vector.
The unit of angular velocity is rad/s.
Rotation with a constant angular velocity is called uniform, while ω = φ/t.
Uniform rotation can be characterized by the period of revolution T, which is understood as the time during which the body makes one revolution, i.e., rotates through an angle of 2π. Since the time interval Δt = Т corresponds to the rotation angle Δφ = 2π, then
(20)
The number of revolutions per unit time ν is obviously equal to:
(21)
The value of ν is measured in hertz (Hz). One hertz is one revolution per second, or 2π rad/s.
The concepts of the period of revolution and the number of revolutions per unit time can also be retained for non-uniform rotation, understanding by the instantaneous value T the time during which the body would complete one revolution if it rotated uniformly with a given instantaneous value of the angular velocity, and by ν, understanding that number revolutions that a body would make per unit of time under similar conditions.
If the angular velocity changes with time, then the rotation is called non-uniform. In this case, enter angular acceleration in the same way as linear acceleration was introduced for rectilinear motion. Angular acceleration is the change in angular velocity per unit of time, calculated as the derivative of the angular velocity with respect to time or the second derivative of the angular displacement with respect to time:
(22)
Just like angular velocity, angular acceleration is a vector quantity. The angular acceleration vector is an axial vector, in the case of accelerated rotation it is directed in the same direction as the angular velocity vector (Fig. 14); in the case of slow rotation, the angular acceleration vector is directed opposite to the angular velocity vector.
In the case of uniformly variable rotational motion, relations similar to formulas (10) and (11), which describe uniformly variable rectilinear motion, take place:
ω = ω 0 ± εt,
.
PHYSICAL VALUES CHARACTERIZING THE MOVEMENT OF A BODY IN A CIRCLE.
1. PERIOD (T) - the period of time during which the body makes one complete revolution.
, where t is the time during which N revolutions are made.
2. FREQUENCY () - the number of revolutions N performed by the body per unit of time.
(hertz)
3. RELATIONSHIP OF PERIOD AND FREQUENCY:
4. MOVEMENT () is directed along the chords.
5. ANGULAR MOVEMENT (angle of rotation).
UNIFORM CIRCULAR MOVEMENT - this is a movement in which the module of speed does not change.
6. LINEAR SPEED (directed tangentially to the circle.
7. ANGULAR VELOCITY 
8. RELATIONSHIP OF LINEAR AND ANGULAR VELOCITIES
The 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 it must be borne in mind that THE ANGULAR VELOCITY OF THESE POINTS IS THE SAME.
9. CENTRIPEAL (normal) ACCELERATION ().
Since when moving along a circle, the direction of the velocity vector is constantly changing, then the movement along the circle occurs with acceleration. If the body moves uniformly along the circle, then it has only centripetal (normal) acceleration, which is directed along the radius to the center of the circle. Acceleration is called normal, since at a given point the acceleration vector is located perpendicular (normally) to the linear velocity vector. .
If the body moves in a circle with a modulo-changing speed, then along with the normal acceleration, which characterizes the change in speed in direction, TANGENTIAL ACCELERATION appears, which characterizes the change in speed modulo (). Directed tangential acceleration tangent to the circle. The total acceleration of the body during non-uniform motion in a circle is determined by the Pythagorean theorem:
RELATIVITY OF MECHANICAL MOTION
When considering the motion of a body relative to different systems reference trajectory, path, speed, displacement are different. For example, a person is sitting in 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 are different from zero. If we consider the motion of a body relative to a moving and stationary frame of reference, then according to the classical law of addition of velocities, the speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of the body relative to the moving frame of reference and the speed of the moving frame of reference relative to the fixed one:
Similarly
SPECIAL CASES OF USING THE LAW OF ADDITION OF VELOCITIES
1) Movement of bodies relative to the Earth
b) bodies move towards each other
2) The movement of bodies relative to each other
a) bodies move in the same direction
b) bodies move in different directions (towards each other)
3) The speed of the body relative to the coast when moving
a) downstream
b) against the current, where is the speed of the body relative to the water, 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 body 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 add the speed of translational and rotational motion vectorially:
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DYNAMICS
NEWTON'S LAWS
NEWTON'S FIRST LAW (LAW OF INERTIA)
There are such frames of reference, in relation to which the body is at rest or moves in a straight line and uniformly, if other bodies do not act on it, or the actions of the bodies are compensated (balanced).
The phenomenon of conservation of the speed of the body in the absence of the action of other bodies on it or when compensating for the action of other bodies is called inertia.
Frames of reference in which Newton's laws are fulfilled are called inertial frames of reference (ISR). IFR includes reference systems connected with the Earth or not having acceleration relative to the Earth. Frames of reference moving with acceleration relative to the Earth are non-inertial, Newton's laws are not fulfilled in them. According to Galileo's classical principle of relativity, all IRFs 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 drawn inside the ISO, it is impossible to determine whether it is at rest or moves in a straight line 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 property of bodies, consisting in the fact that it takes time to change the speed of the body, the speed of the body cannot change instantly. That body, which changes its speed more under the action of the same force, is less inert. The measure of inertia is the mass of the body.
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-directed. If several forces act on the body, then the acceleration tells the body resultant these forces (), which is equal to the vector sum of all forces acting on the body: 
If a body moves with uniform acceleration, 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 one straight line, equal in magnitude and opposite in direction.
Features of the forces arising from the interaction:
1. Forces always appear in pairs.
2 The forces arising from interaction are of the same nature.
3. Forces that do not have a resultant, because they are applied to different bodies.
FORCES IN MECHANICS
GRAVITATION FORCE - the force with which all bodies in the universe are attracted.
THE LAW OF UNIVERSAL GRAVITY: bodies are attracted to 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 gravitation constant), G \u003d 6.67 10 -11, is the mass of the bodies, R is the distance between the bodies, measured between the centers of the bodies. 
GRAVITY FORCE - the force of attraction of bodies to the planet. The force of gravity is calculated by 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 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 the surface of the planet,
The body becomes an artificial satellite if the necessary speed is given to it in the horizontal direction. The speed required for a body to move in a circular orbit around a planet is called first cosmic speed. To obtain a formula for calculating the first cosmic velocity, it must be remembered that all cosmic bodies, including artificial satellites, move under the action of universal gravitation, in addition, velocity is a kinematic quantity, a formula following from Newton's second law Equating the right parts of the formulas, we get: or Given that the body moves in a circle and therefore has centripetal acceleration , we get: or . From here - formula for calculating the first cosmic velocity. Considering that the formula for calculating the first cosmic velocity can be written as: .Similarly, using Newton's second law and the formulas for curvilinear motion, it is possible to determine, for example, the period of a body's orbit.
ELASTIC FORCE - a force acting from the side 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: the elastic force is directly proportional to the elongation: where is the elongation,
Rigidity, . 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 called deformations in which, after the termination of the force, the body acquires its former shape and dimensions.
1. The movement of a body along a circle is called a movement, the trajectory of which is a circle. Moving in a circle, for example, the end of the clock hand, the points of the blade of a rotating turbine, the rotating shaft of the engine, etc.
When moving in a circle, the direction of speed changes continuously. In this case, the modulus of the body's velocity may change, or it may remain unchanged. A movement in which only the direction of the velocity changes, while its modulus remains constant, is called uniform motion of a body in a circle. Under the body this case mean the material point.
2. The motion of a body in a circle is characterized by certain values. 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 complete revolution. The period unit is \([\,T\,] \) = 1 s.
Frequency of circulation\((n) \) - number of full body revolutions in one second: \(n=N/t \) . The unit of frequency is \([\,n\,] \) = 1 s -1 = 1 Hz (hertz). One hertz is the frequency at which the body makes one revolution in one second.
The relationship between frequency and circulation period 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 moving the body from point A to point B, the radius vector will rotate by the angle \(\varphi \) .
The speed of circulation of a body is characterized by angular and linear speed.
Angular velocity \(\omega \) is a physical quantity equal to the ratio of the angle of rotation \(\varphi \) of the radius vector to the time interval during which this rotation occurred: \(\omega=\varphi/t \) . The unit of angular velocity is radians 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 \(2\pi \) . So \(\omega=2\pi/T \) .
Body Linear Velocity\(v \) - the speed with which the body moves along the trajectory. The linear speed with uniform motion along a circle is constant in absolute value, changes in direction and is directed tangentially to the trajectory.
Line speed is equal to the ratio of the path traveled by the body along the trajectory to the time it took to travel this path: \(\vec(v)=l/t \) . In one revolution, the point travels a path equal to the circumference 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 along a circle, the direction of velocity changes, therefore, the difference in velocities is not equal to zero, i.e. the body is moving with acceleration. It is determined by the formula: \(\vec(a)=\frac(\Delta\vec(v))(t) \) and directed in the same way as the velocity change vector. This acceleration is called centripetal acceleration.
centripetal acceleration with uniform motion of the body in a circle - a physical quantity equal to the ratio of the square of the linear velocity 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 absolute value and directed towards the center of the circle.
Part 1
1. When a body moves uniformly in a circle
1) only the modulus 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 at a distance \(R_2=4R_1 \) from the center?
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 revolution of a point along a circle can be calculated by 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 the bicycle wheel increased by 2 times. How has the linear velocity of the wheel rim points changed?
1) increased by 2 times
2) decreased by 2 times
3) increased by 4 times
4) has not changed
6. The linear speed of the points of the helicopter propeller blade has decreased by 4 times. How has their centripetal acceleration changed?
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 along the circumference was increased by 3 times without changing its linear velocity. How has the centripetal acceleration of the body changed?
1) increased by 9 times
2) decreased by 9 times
3) decreased by 3 times
4) increased by 3 times
8. What is the period of revolution of the crankshaft of the engine if it has completed 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 frequency of rotation of the wheel rim point if the period of revolution is 0.05 s?
1) 0.05 Hz
2) 2Hz
3) 20Hz
4) 200Hz
10. The linear speed of the rim point 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.
Set the correspondence between physical quantities in the left column and formulas for their calculation in the right column. In the table under the number of physical
values of the left column, write down the corresponding number of the formula you have chosen from the right column.
PHYSICAL QUANTITY
A) line speed
B) angular velocity
C) frequency of circulation
FORMULA
1) \(1/T \)
2) \(v^2/R \)
3) \(v/R\)
4) \(\omega R \)
5) \(1/n \)
12.
The rotation period of the wheel has increased. How the angular and linear velocities of the wheel rim point 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 you have chosen in the right column.
PHYSICAL QUANTITY
A) angular velocity
B) linear speed
B) centripetal acceleration
CHARACTER OF VALUE CHANGE
1) increased
2) decreased
3) has not changed
Part 2
13. What distance will the wheel rim point cover in 10 s if the wheel revolution frequency is 8 Hz and the wheel radius is 5 m?
Answers
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