Show that centripetal force is also shown by

ANSWER

Centripetal force is the one that compels the body to move in the circle. Mathematically, it is given by the Newton’s Second Law of motion by putting  , where a_c  represents the centripetal acceleration.

Please note that the minus sign indicates that the force is center-seeking whereas radius   is directed away from the center.
Now let’s multiply RHS by 1 = r/r,

Since therefore,

Similarly  v/r = ω , therefore, the above equation becomes,

From equations (1), (3) and (2)

This proves the result.