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. By body in this case we 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 ​\(\omega \) ​ - a physical quantity equal to the ratio of the angle of rotation \(\varphi \) 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. During a time equal to the rotation period, 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 speed 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

When describing the movement of a point along a circle, we will characterize the movement of the point by the angle Δφ , which describes the radius vector of a point over time Δt. Angular displacement in an infinitesimal period of time dt denoted by dφ.

Angular displacement is a vector quantity. The direction of the vector (or ) is determined by the gimlet rule: if you rotate the gimlet (screw with a right-hand thread) in the direction of the point’s movement, the gimlet will move in the direction of the angular displacement vector. In Fig. 14 point M moves clockwise if you look at the plane of movement from below. If you twist the gimlet in this direction, the vector will be directed upward.

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 right-hand thread gimlet rule. However, with the same success one could 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 naturally 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 choosing the positive direction of rotation. In addition, the axial vector does not have an application point. Polar vectors, which we have considered so far, are applied to a moving point. For an axial vector, you can only indicate the direction (axis, axis - Latin) 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.

Motion of a point in a circle

Angular velocity– vector quantity, numerically equal to the angle of rotation per unit time. Angular velocity is usually denoted by the Greek letter ω. 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, just like the angular displacement vector, is an axial vector.

The dimension of angular velocity is rad/s.

Rotation with a constant angular velocity is called uniform, with ω = φ/t.

Uniform rotation can be characterized by the rotation period 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 = T 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 preserved for non-uniform rotation, understanding by the instantaneous value T the time during which the body would make one revolution if it rotated uniformly with a given instantaneous value of angular velocity, and by ν meaning that number revolutions that a body would make per unit time under similar conditions.

If the angular velocity changes with time, then the rotation is called uneven. 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 time, calculated as the derivative of angular velocity with respect to time or the second derivative of 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.

With uniformly variable rotational motion, relations similar to formulas (10) and (11), which describe uniformly variable rectilinear motion, take place:

ω = ω 0 ± εt,

.

Uniform movement around a circle- this is the simplest example. For example, the end of a clock hand moves in a circle around a dial. The speed of a body moving in a circle is called linear speed.

With uniform motion of a body in a circle, the module of the body’s velocity does not change over time, that is, v = const, and only the direction of the velocity vector changes; in this case, there is no change (a r = 0), and the change in the velocity vector in direction is characterized by a quantity called centripetal acceleration() a n or a CS. At each point, the centripetal acceleration vector is directed toward the center of the circle along the radius.

The modulus of centripetal acceleration is equal to

a CS =v 2 / R

Where v is linear speed, R is the radius of the circle

Rice. 1.22. Movement of a body in a circle.

When describing the movement of a body in a circle, we use radius rotation angle– the angle φ through which, during time t, the radius drawn from the center of the circle to the point at which the moving body is located at that moment turns. The rotation angle is measured in radians. equal to the angle between two radii of a 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 equal to

l = 2πR

360 o = 2πR / R = 2π rad.

Hence

1 rad. = 57.2958 o = 57 o 18’

Angular velocity uniform motion of a body in a circle is the value ω, equal to the ratio of the angle of rotation of the radius φ to the period of time during which this rotation is made:

ω = φ / t

The unit of measurement for angular velocity is radian per second [rad/s]. The linear velocity module is determined by the ratio of the length of the traveled path l to the time interval t:

v=l/t

Linear speed with uniform motion around a circle, it is directed along a tangent at a given point on the circle. When a point moves, the length l of the arc of a circle 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.

Circulation period– this is the period of time T during which the body (point) makes one revolution around the circle. Frequency– this is the reciprocal of the period of revolution – the number of revolutions per unit of time (per second). The frequency of circulation is denoted by the letter n.

n=1/T

Over one period, the angle of rotation φ of a point is equal to 2π rad, therefore 2π = ωT, whence

T = 2π/ω

That is, the angular velocity is equal to

ω = 2π / T = 2πn

Centripetal acceleration can be expressed in terms of period T and circulation frequency n:

a CS = (4π 2 R) / T 2 = 4π 2 Rn 2

Circular movement.

1.Uniform movement in a circle

2. Angular speed of rotational motion.

3. Rotation period.

4. Rotation speed.

5. Relationship between linear speed and angular speed.

6.Centripetal acceleration.

7. Equally alternating movement in a circle.

8. Angular acceleration in uniform circular motion.

9.Tangential acceleration.

10. Law of uniformly accelerated motion in a circle.

11. Average angular velocity in uniformly accelerated motion in a circle.

12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

1.Uniform movement around a circle– movement in which a material point passes equal segments of a circular arc in equal time intervals, i.e. the point moves in a circle with a constant absolute speed. In this case, the speed is equal to the ratio of the arc of a circle traversed by the point to the time of movement, i.e.

and is called the linear speed of movement in a circle.

As in curvilinear motion, the velocity vector is directed tangentially to the circle in the direction of motion (Fig. 25).

2. Angular velocity in uniform circular motion– ratio of the radius rotation angle to the rotation time:

In uniform circular motion, the angular velocity is constant. In the SI system, angular velocity is measured in (rad/s). One radian - a rad is the central angle subtending an arc of a circle with a length equal to the radius. A full angle contains radians, i.e. per revolution the radius rotates by an angle of radians.

3. Rotation period– time interval T during which a material point makes one full revolution. In the SI system, the period is measured in seconds.

4. Rotation frequency– the number of revolutions made in one second. In the SI system, frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is completed in one second. It's easy to imagine that

If during time t a point makes n revolutions around a circle then .

Knowing the period and frequency of rotation, the angular velocity can be calculated using the formula:

5 Relationship between linear speed and angular speed. The length of an arc of a circle is equal to where is the central angle, expressed in radians, the radius of the circle subtending the arc. Now we write the linear speed in the form

It is often convenient to use the formulas: or Angular velocity is often called cyclic frequency, and frequency is called linear frequency.

6. Centripetal acceleration. In uniform motion around a circle, the velocity module remains unchanged, but its direction continuously changes (Fig. 26). This means that a body moving uniformly in a circle experiences acceleration, which is directed towards the center and is called centripetal acceleration.

Let a distance travel equal to an arc of a circle in a period of time. Let's move the vector, leaving it parallel to itself, so that its beginning coincides with the beginning of the vector at point B. The modulus of change in speed is equal to , and the modulus of centripetal acceleration is equal

In Fig. 26, the triangles AOB and DVS are isosceles and the angles at the vertices O and B are equal, as are the angles with mutually perpendicular sides AO and OB. This means that the triangles AOB and DVS are similar. Therefore, if, that is, the time interval takes arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. . Therefore, we can write Considering that VD = , OA = R we obtain Multiplying both sides of the last equality by , we further obtain the expression for the modulus of centripetal acceleration in uniform motion in a circle: . Considering that we get two frequently used formulas:

So, in uniform motion around a circle, the centripetal acceleration is constant in magnitude.

It is easy to understand that in the limit at , angle . This means that the angles at the base of the DS of the ICE triangle tend to the value , and the speed change vector becomes perpendicular to the speed vector, i.e. directed radially towards the center of the circle.

7. Equally alternating circular motion– circular motion in which the angular velocity changes by the same amount over equal time intervals.

8. Angular acceleration in uniform circular motion– the ratio of the change in angular velocity to the time interval during which this change occurred, i.e.

where the initial value of angular velocity, the final value of angular velocity, angular acceleration, in the SI system is measured in . From the last equality we obtain formulas for calculating the angular velocity

And if .

Multiplying both sides of these equalities by and taking into account that , is the tangential acceleration, i.e. acceleration directed tangentially to the circle, we obtain formulas for calculating linear speed:

And if .

9. Tangential acceleration numerically equal to the change in speed per unit time and directed along the tangent to the circle. If >0, >0, then the motion is uniformly accelerated. If<0 и <0 – движение.

10. Law of uniformly accelerated motion in a circle. The path traveled around a circle in time in uniformly accelerated motion is calculated by the formula:

Substituting , , and reducing by , we obtain the law of uniformly accelerated motion in a circle:

Or if.

If the movement is uniformly slow, i.e.<0, то

11.Total acceleration in uniformly accelerated circular motion. In uniformly accelerated motion in a circle, centripetal acceleration increases over time, because Due to tangential acceleration, linear speed increases. Very often, centripetal acceleration is called normal and is denoted as. Since the total acceleration at a given moment is determined by the Pythagorean theorem (Fig. 27).

12. Average angular velocity in uniformly accelerated motion in a circle. The average linear speed in uniformly accelerated motion in a circle is equal to . Substituting here and and reducing by we get

If, then.

12. Formulas establishing the relationship between angular velocity, angular acceleration and angle of rotation in uniformly accelerated motion in a circle.

Substituting the quantities , , , , into the formula

and reducing by , we get

Lecture-4. Dynamics.

1. Dynamics

2. Interaction of bodies.

3. Inertia. The principle of inertia.

4. Newton's first law.

5. Free material point.

6. Inertial reference system.

7. Non-inertial reference system.

8. Galileo's principle of relativity.

9. Galilean transformations.

11. Addition of forces.

13. Density of substances.

14. Center of mass.

15. Newton's second law.

16. Unit of force.

17. Newton's third law

1. Dynamics there is a branch of mechanics that studies mechanical motion, depending on the forces that cause a change in this motion.

2.Interactions of bodies. Bodies can interact both in direct contact and at a distance through a special type of matter called a physical field.

For example, all bodies are attracted to each other and this attraction is carried out through the gravitational field, and the forces of attraction are called gravitational.

Bodies carrying an electric charge interact through an electric field. Electric currents interact through a magnetic field. These forces are called electromagnetic.

Elementary particles interact through nuclear fields and these forces are called nuclear.

3.Inertia. In the 4th century. BC e. The Greek philosopher Aristotle argued that the cause of the movement of a body is the force acting from another body or bodies. At the same time, according to Aristotle’s movement, a constant force imparts a constant speed to the body and, with the cessation of the action of the force, the movement ceases.

In the 16th century Italian physicist Galileo Galilei, conducting experiments with bodies rolling down an inclined plane and with falling bodies, showed that a constant force (in this case, the weight of a body) imparts acceleration to the body.

So, based on experiments, Galileo showed that force is the cause of the acceleration of bodies. Let us present Galileo's reasoning. Let a very smooth ball roll along a smooth horizontal plane. If nothing interferes with the ball, then it can roll for as long as desired. If a thin layer of sand is poured on the path of the ball, it will stop very soon, because it was affected by the frictional force of the sand.

So Galileo came to the formulation of the principle of inertia, according to which a material body maintains a state of rest or uniform linear motion if no external forces act on it. This property of matter is often called inertia, and the movement of a body without external influences is called motion by inertia.

4. Newton's first law. In 1687, based on Galileo's principle of inertia, Newton formulated the first law of dynamics - Newton's first law:

A material point (body) is in a state of rest or uniform linear motion if other bodies do not act on it, or the forces acting from other bodies are balanced, i.e. compensated.

5.Free material point- a material point that is not affected by other bodies. Sometimes they say - an isolated material point.

6. Inertial reference system (IRS)– a reference system relative to which an isolated material point moves rectilinearly and uniformly, or is at rest.

Any reference system that moves uniformly and rectilinearly relative to the ISO is inertial,

Let us give another formulation of Newton's first law: There are reference systems relative to which a free material point moves rectilinearly and uniformly, or is at rest. Such reference systems are called inertial. Newton's first law is often called the law of inertia.

Newton's first law can also be given the following formulation: every material body resists a change in its speed. This property of matter is called inertia.

We encounter manifestations of this law every day in urban transport. When the bus suddenly picks up speed, we are pressed against the back of the seat. When the bus slows down, our body skids in the direction of the bus.

7. Non-inertial reference system – a reference system that moves unevenly relative to the ISO.

A body that, relative to the ISO, is in a state of rest or uniform linear motion. It moves unevenly relative to a non-inertial reference frame.

Any rotating reference system is a non-inertial reference system, because in this system the body experiences centripetal acceleration.

There are no bodies in nature or technology that could serve as ISOs. For example, the Earth rotates around its axis and any body on its surface experiences centripetal acceleration. However, for fairly short periods of time, the reference system associated with the Earth’s surface can, to some approximation, be considered ISO.

8.Galileo's principle of relativity. ISO can be as much salt as you like. Therefore, the question arises: what do the same mechanical phenomena look like in different ISOs? Is it possible, using mechanical phenomena, to detect the movement of the ISO in which they are observed.

The answer to these questions is given by the principle of relativity of classical mechanics, discovered by Galileo.

The meaning of the principle of relativity of classical mechanics is the statement: all mechanical phenomena proceed exactly the same way in all inertial frames of reference.

This principle can be formulated as follows: all laws of classical mechanics are expressed by the same mathematical formulas. In other words, no mechanical experiments will help us detect the movement of the ISO. This means that trying to detect ISO movement is meaningless.

We encountered the manifestation of the principle of relativity while traveling on trains. At the moment when our train is standing at the station, and the train standing on the adjacent track slowly begins to move, then in the first moments it seems to us that our train is moving. But it also happens the other way around, when our train smoothly picks up speed, it seems to us that the neighboring train has started moving.

In the above example, the principle of relativity manifests itself over small time intervals. As the speed increases, we begin to feel shocks and swaying of the car, i.e. our reference system becomes non-inertial.

So, trying to detect ISO movement is pointless. Consequently, it is absolutely indifferent which ISO is considered stationary and which is moving.

9. Galilean transformations. Let two ISOs move relative to each other with a speed. In accordance with the principle of relativity, we can assume that the ISO K is stationary, and the ISO moves relatively at a speed. For simplicity, we assume that the corresponding coordinate axes of the systems and are parallel, and the axes and coincide. Let the systems coincide at the moment of beginning and the movement occurs along the axes and , i.e. (Fig.28)

Circular motion is the simplest case of curvilinear motion of a body. When a body moves around a certain point, along with the displacement vector it is convenient to enter the angular displacement ∆ φ (angle of rotation relative to the center of the circle), measured in radians.

Knowing the angular displacement, you can calculate the length of the circular arc (path) that the body has traversed.

∆ l = R ∆ φ

If the angle of rotation is small, then ∆ l ≈ ∆ s.

Let us illustrate what has been said:

Angular velocity

With curvilinear motion, the concept of angular velocity ω is introduced, that is, the rate of change in the angle of rotation.

Definition. Angular velocity

The angular velocity at a given point of the trajectory is the limit of the ratio of the angular displacement ∆ φ to the time interval ∆ t during which it occurred. ∆ t → 0 .

ω = ∆ φ ∆ t , ∆ t → 0 .

The unit of measurement for angular velocity is radian per second (r a d s).

There is a relationship between the angular and linear speeds of a body when moving in a circle. Formula for finding angular velocity:

With uniform motion in a circle, the velocities v and ω remain unchanged. Only the direction of the linear velocity vector changes.

In this case, uniform motion in a circle affects the body by centripetal, or normal acceleration, directed along the radius of the circle to its center.

a n = ∆ v → ∆ t , ∆ t → 0

The modulus of centripetal acceleration can be calculated using the formula:

a n = v 2 R = ω 2 R

Let us prove these relations.

Let's consider how the vector v → changes over a short period of time ∆ t. ∆ v → = v B → - v A → .

At points A and B, the velocity vector is directed tangentially to the circle, while the velocity modules at both points are the same.

By definition of acceleration:

a → = ∆ v → ∆ t , ∆ t → 0

Let's look at the picture:

Triangles OAB and BCD are similar. It follows from this that O A A B = B C C D .

If the value of the angle ∆ φ is small, the distance A B = ∆ s ≈ v · ∆ t. Taking into account that O A = R and C D = ∆ v for the similar triangles considered above, we obtain:

R v ∆ t = v ∆ v or ∆ v ∆ t = v 2 R

When ∆ φ → 0, the direction of the vector ∆ v → = v B → - v A → approaches the direction to the center of the circle. Assuming that ∆ t → 0, we obtain:

a → = a n → = ∆ v → ∆ t ; ∆ t → 0 ; a n → = v 2 R .

With uniform motion around a circle, the acceleration modulus remains constant, and the direction of the vector changes with time, maintaining orientation to the center of the circle. That is why this acceleration is called centripetal: the vector at any moment of time is directed towards the center of the circle.

Writing centripetal acceleration in vector form looks like this:

a n → = - ω 2 R → .

Here R → is the radius vector of a point on a circle with its origin at its center.

In general, acceleration when moving in a circle consists of two components - normal and tangential.

Let us consider the case when a body moves unevenly around a circle. Let us introduce the concept of tangential (tangential) acceleration. Its direction coincides with the direction of the linear velocity of the body and at each point of the circle is directed tangent to it.

a τ = ∆ v τ ∆ t ; ∆ t → 0

Here ∆ v τ = v 2 - v 1 - change in velocity module over the interval ∆ t

The direction of the total acceleration is determined by the vector sum of the normal and tangential accelerations.

Circular motion in a plane can be described using two coordinates: x and y. At each moment of time, the speed of the body can be decomposed into components v x and v y.

If the motion is uniform, the quantities v x and v y as well as the corresponding coordinates will change in time according to a harmonic law with a period T = 2 π R v = 2 π ω

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