Circular Motion

Circular motion

Circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform or non-uniform.

Since the object’s velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation.

Uniform circular motion: It describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times.

Formulas for uniform circular motion:

If the period for one rotation is T, the angular rate of rotation (angular velocity), ω is:

Ω = 2π / T

and the units are radians/sec

The speed of the object traveling the circle is:

v = 2 π r / T = ω r

The angle θ swept out in a time t is:

Θ = 2 π t / T = ω t

The acceleration due to change in the direction is:

a = v2 / r = ω2 r

Velocity

For a path of radius r, when an angle θ is swept out, the distance travelled on the periphery of the orbit is

s = r θ.

Therefore, the speed of travel around the orbit is

v = r dθ / dt = r ω

Acceleration

a= v dθ / dt = v ω = v2 / r

Torque

Torque is the tendency of a force to rotate an object about an axis.

Mathematically, torque is defined as the cross product of the lever-arm distance and force, which tends to produce rotation.

The magnitude of torque depends on three quantities: the force applied, the length of the lever arm connecting the axis to the point of force application, and the angle between the force vector and the lever arm.

T = r F sinθ

where

r is the length (or magnitude) of the lever arm vector,

F is the force vector

θ is the angle between the force vector and the lever arm vector.

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