Circular movement. Equation of motion in a circle. Angular velocity. Normal = centripetal acceleration. Period, frequency of circulation (rotation). Relationship between linear and angular velocity Velocity vector for circular motion

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

Such movement is performed by the 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 speed during such movement continuously changes.

The speed of movement of a body at any point on a curvilinear trajectory is directed tangentially to the trajectory at that point. You can verify this by observing the operation of a disk-shaped sharpener: pressing the end of a steel rod against a rotating stone, you can see hot particles coming off the stone. These particles fly at the speed they had at the moment they left the stone. The direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. The splashes from the wheels of a skidding car also move tangentially to the circle.

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

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

Both the magnitude and direction of acceleration must depend on the shape of the curved trajectory. However, there is no need to consider each of its countless forms. Having imagined each section as a separate circle with a certain radius, the problem of finding acceleration during curvilinear uniform motion will be reduced to finding acceleration during uniform motion of a body in a circle.

Uniform circular motion is characterized by the period and frequency of revolution.

The time it takes 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 by the speed of movement:

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

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

When a body moves uniformly around a circle, the acceleration at any point 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 linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.

Angular velocity

Let's choose a point on the circle 1 . Let's build a radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.

Period and frequency

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

Rotation frequency is the number of revolutions per second.

Frequency and period are interrelated by the relationship

Relationship with angular velocity

Linear speed

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 under a grinding machine move, repeating the direction of instantaneous speed.

Consider a point on a circle that makes one revolution, the time spent is the period T The path that a point travels is the circumference.

Centripetal acceleration

When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.

Using the previous formulas, we can derive the following relationships

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

The law of addition of speeds 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: 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 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 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 with the disk around its axis, then such a force is the friction force. If the force stops its action, 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 speed is equal to

Now let's move to a stationary system connected to the ground. The total acceleration of point A will remain the same both in magnitude and direction, since when moving from one inertial reference system 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.

Movement of a body in a circle with a constant absolute speed- this is a movement in which a body describes identical arcs at 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 time Δ t body moving from a point A exactly IN, 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 rotates by an angle Δ φ . The angle is expressed in radians.

The speed \(~\vec \upsilon\) of a body's movement along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval Δ t for which this arc is completed:

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

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

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

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

With uniform motion in a circle, the angular velocity and the linear velocity module are constant quantities: 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 angular coordinate is φ 0 , and at time t it is equal φ , then the rotation angle Δ φ radius vector for 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 obtain\[~\omega = \frac(l)(R \Delta t) = \frac(\upsilon)(R) \Rightarrow\]

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

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

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

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

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

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

Magnitude ν , the inverse of the period, showing how many revolutions a body makes per unit time, is called rotation 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 secondary school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 18-19.