Consider an object  A  moving in a circular motion as shown in the figure above.

circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation

from the diagram above

If the radius of the circle is r and the object is moving with uniform speed of v, the linear velocity is    

Relationship between angular and linear velocity.

From the above equation

The linear velocity of an object moving in a circular motion is the products of the Angular velocity of the object and the radius of the circular path  in radians.

Example

An object is attached to the blade of a ceiling fan. If the object is at 1m from the centre of the fan , and the object makes 9 complete revolution in 3 seconds. What is the Angular and linear velocity of the object ?

Solution

The object moves through an angle equal to 9 x 2  radians in 3 seconds (360=2 )

Acceleration of simple harmonic Motion and circular motion

An object moving in circular motion experiences a constant acceleration directed towards the centre of the motion.

The centripetal acceleration is given by

             As proved earlier  , where r is the radius of the circle in circular motion 

Angular acceleration and linear acceleration of circular motion

If the angular velocity of an object moving in circular motion changes from     then the angular acceleration  is given by

   Linear acceleration a of an object moving in a circular motion is the products of the angular acceleration  and the radius .

General observation of simple harmonic motion

1. The speed is at maximum at the centre position c and zero at A where the displacement is maximum.
2. At centre C the body is at equilibrium , the displacement is zero. The speed of the body is maximum and acceleration is zero.
3. The period of the motion is given by    . This means the period is independent of the amplitude and mass of the body.

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