Formula for acceleration in circular motion. Uniform movement around a circle

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:

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

The number of revolutions per unit time ν is obviously equal to:

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:

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.

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 earth. 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.

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