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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