Q39 of 111 Page 252

A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.


given,


Radius of the surfaces = R


Radius of the ball = r


Mass of the ball = m


Let angular amplitude be


Torque on the ball




Moment of inertia of ball


( for spherical surface)


(parallel axis theorem)



Angular acceleration of a ball




Angular acceleration about the center of the surface


(effective distance R-r)


(for smaller angles )


Linear acceleration




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