Arc movement. Uniform circular motion. Period and frequency

the beauty

Movement of a body in a circle with a constant modulo speed- this is a movement in which the body describes the same arcs for any equal intervals of time.

The position of the body on the circle 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 a point BUT exactly AT, moves \(~\Delta \vec r\) 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 the movement of the body along the trajectory (circle) is directed along the tangent to the trajectory. It is called linear speed. The linear velocity modulus is equal to the ratio of the length of the circular arc l to the time interval Δ t for which this arc is passed:

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

A 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:

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

The SI unit of angular velocity is the radian per second (rad/s).

With uniform motion in a circle, the angular velocity and the linear velocity modulus 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 time t 0 = 0 the angular coordinate is φ 0 , and at time t it is equal to φ , then the rotation angle 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 we can get kinematic equation of motion of a material point along a circle:

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

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

\(~\upsilon = \omega R\) - 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 traverses the path \(~l = 2 \pi R\). Consequently,

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

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

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

Consequently,

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

Literature

Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vykhavanne, 2004. - C. 18-19.

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, a uniform movement 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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Since the linear 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 a grinder move, repeating the direction of instantaneous speed.

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.

Among various kinds curvilinear motion is of particular interest uniform motion of a body in a circle. This is the simplest form of curvilinear motion. At the same time, any complex curvilinear motion of a body in a sufficiently small section of its trajectory can be approximately considered as uniform motion along a circle.

Such a movement is made by points of rotating wheels, turbine rotors, artificial satellites rotating in orbits, etc. With uniform motion in a circle, the numerical value of the speed remains constant. However, the direction of the velocity during such a movement is constantly changing.

The speed of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at this point. This can be seen by observing the work of a disc-shaped grindstone: pressing the end of a steel rod to a rotating stone, you can see hot particles coming off the stone. These particles fly at the same speed that they had at the moment of separation from the stone. The direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. Sprays from the wheels of a skidding car also move tangentially to the circle.

Thus, the instantaneous velocity of the body at different points of the curvilinear trajectory has different directions, while the modulus of velocity can either be the same everywhere or change from point to point. But even if the modulus of speed does not change, it still cannot be considered constant. After all, speed is a vector quantity, and for vector quantities, the modulus and direction are equally important. That's why curvilinear motion is always accelerated, even if the modulus of speed is constant.

Curvilinear motion can change the speed modulus and its direction. Curvilinear motion, in which the modulus of speed remains constant, is called uniform curvilinear motion. Acceleration during such movement is associated only with a change in the direction of the velocity vector.

Both the modulus and the direction of acceleration must depend on the shape of the curved trajectory. However, it is not necessary to consider each of its myriad forms. Representing each section as a separate circle with a certain radius, the problem of finding acceleration in a curvilinear uniform motion will be reduced to finding acceleration in a body moving uniformly along a circle.

Uniform motion in a circle is characterized by a period and frequency of circulation.

The time it takes for a body to make one revolution is called circulation period.

With uniform motion in a circle, the period of revolution is determined by dividing the distance traveled, i.e., the circumference of the circle by the speed of movement:

The reciprocal of a period is called circulation frequency, denoted by the letter ν . Number of revolutions per unit time ν called circulation frequency:

Due to the continuous change in the direction of speed, a body moving in a circle has an acceleration that characterizes the speed of change in its direction, the numerical value of the speed in this case does not change.

With a uniform motion of a body along a circle, the acceleration at any point in it is always directed perpendicular to the speed of movement along the radius of the circle to its center and is called centripetal acceleration.

To find its value, consider the ratio of the change in the velocity vector to the time interval during which this change occurred. Since the angle is very small, we have

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

Angular velocity

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

Consider a point on a circle that makes one revolution, the time that is spent - this is the period T. The path traveled by a point is the circumference of a 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

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 v A and v B respectively. Acceleration is the change in speed per unit of time. Let's find the difference of vectors.