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
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
a= v dθ / dt = v ω = v2 / r
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
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.