Circular movement. Circular motion equation

Topics of the USE codifier: movement in a circle with a constant modulo speed, centripetal acceleration.

Uniform circular motion is a fairly simple example of motion with an acceleration vector that depends on time.

Let the point rotate on a circle of radius . The speed of a point is constant modulo and equal to . The speed is called linear speed points.

Period of circulation is the time for one complete revolution. For the period, we have an obvious formula:

. (1)

Frequency of circulation is the reciprocal of the period:

The frequency indicates how many complete revolutions the point makes per second. The frequency is measured in rpm (revolutions per second).

Let, for example, . This means that during the time the point makes one complete
turnover. The frequency in this case is equal to: about / s; The point makes 10 complete revolutions per second.

Angular velocity.

Consider the uniform rotation of a point in the Cartesian coordinate system. Let's place the origin of coordinates in the center of the circle (Fig. 1).

Rice. 1. Uniform circular motion

Let be the initial position of the point; in other words, for , the point had coordinates . Let the point turn through an angle in time and take the position .

The ratio of the angle of rotation to time is called angular velocity point rotation:

. (2)

Angle is usually measured in radians, so angular velocity is measured in rad/s. For a time equal to the period of rotation, the point rotates through an angle. That's why

. (3)

Comparing formulas (1) and (3), we obtain the relationship between linear and angular velocities:

. (4)

The law of motion.

Let us now find the dependence of the coordinates of the rotating point on time. We see from Fig. 1 that

But from formula (2) we have: . Hence,

. (5)

Formulas (5) are the solution to the main problem of mechanics for the uniform motion of a point along a circle.

centripetal acceleration.

Now we are interested in the acceleration of the rotating point. It can be found by differentiating relations (5) twice:

Taking into account formulas (5), we have:

(6)

The resulting formulas (6) can be written as a single vector equality:

(7)

where is the radius vector of the rotating point.

We see that the acceleration vector is directed opposite to the radius vector, i.e., towards the center of the circle (see Fig. 1). Therefore, the acceleration of a point moving uniformly in a circle is called centripetal.

In addition, from formula (7) we obtain an expression for the modulus of centripetal acceleration:

(8)

We express the angular velocity from (4)

and substitute into (8) . Let's get one more formula for centripetal acceleration.

In this lesson, we will consider curvilinear motion, namely the uniform motion of a body in a circle. We will learn what linear speed is, centripetal acceleration when a body moves in a circle. We also introduce quantities that characterize the rotational motion (rotation period, rotation frequency, angular velocity), and connect these quantities with each other.

By uniform motion in a circle is understood that the body rotates through the same angle for any identical period of time (see Fig. 6).

Rice. 6. Uniform circular motion

That is, the module of instantaneous speed does not change:

This speed is called linear.

Although the modulus of the speed does not change, the direction of the speed changes continuously. Consider the velocity vectors at the points A And B(see Fig. 7). They are directed in different directions, so they are not equal. If subtracted from the speed at the point B point speed A, we get a vector .

Rice. 7. Velocity vectors

The ratio of the change in speed () to the time during which this change occurred () is acceleration.

Therefore, any curvilinear motion is accelerated.

If we consider the velocity triangle obtained in Figure 7, then with a very close arrangement of points A And B to each other, the angle (α) between the velocity vectors will be close to zero:

It is also known that this triangle is isosceles, so the modules of velocities are equal (uniform motion):

Therefore, both angles at the base of this triangle are indefinitely close to:

This means that the acceleration that is directed along the vector is actually perpendicular to the tangent. It is known that a line in a circle perpendicular to a tangent is a radius, so acceleration is directed along the radius towards the center of the circle. This acceleration is called centripetal.

Figure 8 shows the triangle of velocities discussed earlier and an isosceles triangle (two sides are the radii of a circle). These triangles are similar, since they have equal angles formed by mutually perpendicular lines (the radius, like the vector, is perpendicular to the tangent).

Rice. 8. Illustration for the derivation of the centripetal acceleration formula

Line segment AB is move(). We are considering uniform circular motion, so:

Substitute the resulting expression for AB into the triangle similarity formula:

The concepts of "linear speed", "acceleration", "coordinate" are not enough to describe the movement along a curved trajectory. Therefore, it is necessary to introduce quantities characterizing the rotational motion.

1. The rotation period (T ) is called the time of one complete revolution. It is measured in SI units in seconds.

Examples of periods: The Earth rotates around its axis in 24 hours (), and around the Sun - in 1 year ().

Formula for calculating the period:

where is the total rotation time; - number of revolutions.

2. Rotation frequency (n ) - the number of revolutions that the body makes per unit of time. It is measured in SI units in reciprocal seconds.

Formula for finding the frequency:

where is the total rotation time; - number of revolutions

Frequency and period are inversely proportional:

3. angular velocity () called the ratio of the change in the angle at which the body turned to the time during which this turn occurred. It is measured in SI units in radians divided by seconds.

Formula for finding the angular velocity:

where is the change in angle; is the time it took for the turn to take place.

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 introduce the angular displacement ∆ φ (the angle of rotation relative to the center of the circle), measured in radians.

Knowing the angular displacement, it is possible to calculate the length of the circular arc (path) that the body has passed.

∆ l = R ∆ φ

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

Let's 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 measure for angular velocity is radians per second (r a d s).

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

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

In this case, uniform motion along a circle on the body is affected by 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 by 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 small 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 \u003d R and C D \u003d ∆ v for the similar triangles considered above, 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. Assuming 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 module remains constant, and the direction of the vector changes with time, while maintaining orientation to the center of the circle. That is why this acceleration is called centripetal: the vector at any time is directed towards the center of the circle.

The record of centripetal acceleration in vector form is as follows:

a n → = - ω 2 R → .

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

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

Consider the case when the body moves along the circle non-uniformly. 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 tangentially to it.

a τ = ∆ v τ ∆ t ; ∆t → 0

Here ∆ v τ \u003d v 2 - v 1 is the change in the velocity module over the interval ∆ t

The direction of full acceleration is determined by the vector sum of 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 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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