Question

A uniform ring of mass m, radius r placed on a smooth horizontal surface is rotated about its central axis with constant angular velocity . Calculate tension in the ring.

Answer 1

Mass per unit length of ring = M/(2*\Pi*R)

Mass of the dx element = dm = dx *M/(2*\Pi*R)

For equilibrium,

2T sin\Theta= dm * v2/ R

Since,\Theta= small,sin\Theta=\Theta

2T\Theta=dx *M/(2\PiR)* v2 / R

2T (dx/R) =dx *M/(2\PiR)* v2?/ R Tension in the ring. T= Mv2/ (4\PiR)