Question

A particle of mass 1 g executes an oscillatory motion on the concave surface of a spherical dish of radius 2 m placed on a horizontal plane, as in the figure below. If the motion of the particle begins from a point on the dish at a height of 1 cm from the horizontal plane and coefficient of friction is 0.01, find the total distance covered by the particle before coming to rest.

Answer 1

In the fig.,CA = CO = 200 cm

OB = 1 cm

BC = CO - OB = 200 - 1 = 199 cm

In ΔABC, cosθ = BC/AC = 199/200

Treating the dish as an inclined plane, and using the principle of energy conservation, i.e.,

Loss in P.E. = work done against friction

or, mgh = μR x S = μ mg cosθ x S

or, S = {mgh}/{μ mg cosθ} = {h}/{μ cosθ}

S = {1/100}/{0.01 x 199/200} = 200/199 = 1.005 m