Equation of motion in a circle. Uniform circular motion

The movement of a body in a circle with a constant modulo velocity is a movement in which the body describes the same arcs at any equal time intervals.

The position of the body on the circumference is determined radius vector\ (~ \ vec r \) drawn from the center of the circle. The modulus of the radius vector is equal to the radius of the circle R(fig. 1).

During the time Δ t body moving from point A exactly V, makes a displacement \ (~ \ Delta \ vec r \) equal to the chord AB, and travels a path equal to the length of the arc l.

The radius vector is rotated by an angle Δ φ ... The angle is expressed in radians.

The speed \ (~ \ vec \ upsilon \) of body movement along the trajectory (circle) is directed tangentially to the trajectory. It is called linear velocity... The linear velocity modulus is equal to the ratio of the length of the arc of a circle l to the time interval Δ t for which this arc is passed:

\ (~ \ upsilon = \ frac (l) (\ Delta t). \)

Scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the time interval during which this rotation occurred, is called angular velocity:

\ (~ \ omega = \ frac (\ Delta \ varphi) (\ Delta t). \)

In SI, the unit of angular velocity is radians per second (rad / s).

With uniform movement around a circle, the angular velocity and the modulus of the linear velocity are constant values: ω = const; υ = const.

The position of the body can be determined if the modulus of the radius vector \ (~ \ vec r \) and the angle φ which it composes with the axis Ox(angular coordinate). If at the initial moment of time t 0 = 0 the angular coordinate is φ 0, and at the moment of time t it is equal φ , then the angle of rotation Δ φ radius-vector in time \ (~ \ Delta t = t - t_0 = t \) is equal to \ (~ \ Delta \ varphi = \ varphi - \ varphi_0 \). Then from the last formula one can obtain kinematic equation of motion material point circumferentially:

\ (~ \ varphi = \ varphi_0 + \ omega t. \)

It allows you to determine the position of the body at any time t... Considering that \ (~ \ Delta \ varphi = \ frac (l) (R) \), we get \ [~ \ omega = \ frac (l) (R \ Delta t) = \ frac (\ upsilon) (R) \ Rightarrow \]

\ (~ \ upsilon = \ omega R \) - the formula for the relationship between linear and angular velocity.

Time interval Τ , during which the body makes one complete revolution, is called rotation period:

\ (~ T = \ frac (\ Delta t) (N), \)

where N- the number of revolutions made by the body during the time Δ t.

During the time Δ t = Τ the body goes along the path \ (~ l = 2 \ pi R \). Hence,

\ (~ \ upsilon = \ frac (2 \ pi R) (T); \ \ omega = \ frac (2 \ pi) (T). \)

The magnitude ν , the inverse of the period, showing how many revolutions the body makes per unit of time, is called rotational speed:

\ (~ \ nu = \ frac (1) (T) = \ frac (N) (\ Delta t). \)

Hence,

\ (~ \ upsilon = 2 \ pi \ nu R; \ \ omega = 2 \ pi \ nu. \)

Literature

Aksenovich L.A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing the receipt of obs. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Minsk: Adukatsya i vyhavanne, 2004. - pp. 18-19.

Since the linear speed uniformly changes direction, the movement along a circle cannot be called uniform, it is equally accelerated.

Angular velocity

Choose a point on the circle 1 ... Let's build a radius. In 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 of time.

Period and frequency

Rotation period T- this is the time during which the body makes one revolution.

Rotation speed is the number of revolutions per second.

Frequency and period are interrelated by the ratio

Angular Velocity Relationship

Linear Velocity

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 a grinder move in the same direction as the instantaneous speed.

Consider a point on a circle that makes one revolution, the time it takes is a period T. The path that the point overcomes is the circumference.

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, the following relations can be derived

Points lying on one straight line outgoing from the center of the circle (for example, these can be points that lie on the spoke of a wheel) will have the same angular velocity, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The further the point is from the center, the faster it will move.

The law of addition of velocities is also valid for rotary motion... If the movement of a body or a frame of reference is not uniform, then the law is applied for 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 person's movement speed.

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 equatorial plane and the direction from the center of the Earth to a point on its surface.

According to Newton's second law, force is the cause of any acceleration. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration can 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 ceases to act, then the body will 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 us go over to a stationary system connected to the earth. The total acceleration of point A will remain the same both in magnitude and in direction, since when passing from one inertial frame of reference to another, the acceleration does not change. 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 circumferentially Is the simplest example. For example, the end of the watch hand moves along the circle along the dial. The speed of movement of the body in a circle is called line speed.

With uniform movement of the body around the circumference, the modulus 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 (a r = 0), and the change in the velocity vector in the direction is characterized by a quantity 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 centripetal acceleration modulus is

a CA = 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 movement of a body in a circle, it is used radius rotation angle- the angle φ, by which, in time t, the radius turns from the center of the circle to the point at which the moving body is at this moment. The angle of rotation 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 is equal to

l = 2πR

360 о = 2πR / R = 2π rad.

Hence

1 glad. = 57.2958 o = 57 o 18 '

Angular velocity uniform movement of the body along the circumference is the value of ω, equal to the ratio of the angle of rotation of the radius φ to the time interval during which this rotation was 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 length of the traversed path l to the time interval t:

v = l / t

Linear Velocity with uniform movement along a circle, it is directed tangentially at a given point of the circle. When a 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. Call frequency Is the reciprocal of the period of revolution - the number of revolutions per unit of time (per second). The frequency of the call is indicated by the letter n.

n = 1 / T

In one period, the angle of rotation φ of a 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 circulation n:

a CS = (4π 2 R) / T 2 = 4π 2 Rn 2

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

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

∆ l = R ∆ φ

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

Let us illustrate what has been said:

Angular velocity

In 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 measure for angular velocity is radians per second (rad s).

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

With uniform motion around the circumference, the speeds v and ω remain unchanged. Only the direction of the linear velocity vector changes.

In this case, a uniform movement around the circumference of the body acts centripetal, or normal acceleration directed along the radius of the circle to its center.

a n = ∆ v → ∆ t, ∆ t → 0

The centripetal acceleration module can be calculated using the formula:

a n = v 2 R = ω 2 R

Let us prove these relations.

Let us consider how the vector v → changes in a small time interval ∆ 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 take a 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 above similar triangles we get:

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. Taking that ∆ t → 0, we get:

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

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

Recording 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 the origin at its center.

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

Consider the case when the body moves unevenly around the circle. Let's 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 tangentially to it.

a τ = ∆ v τ ∆ t; ∆ t → 0

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

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

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 values ​​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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