Question

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. In each case, the ball will _____________

(a) Reach the bottom at the same time

(b) Will take longer time to roll down one plane

(c) Reach the bottom at an unpredictable time

(d) Reach the bottom at the same time and keeps rolling

The question was asked in final exam.

My question is based upon Angular Momentum in chapter System of Particles and Rotational Motion of Engineering Physics I

Answer 1

The correct option is (b) Will take longer time to roll down one plane

The explanation: Acceleration of the rolling sphere,

a=gsinθ/((1+k^2/R^2))

Velocity of the sphere at the bottom of the inclined plane,

v=√(2gh/((1+k^2/R^2)))

The sphere will take a longer time to roll down one plane than the other. It will take a larger time in case if the plane with smaller inclination because the acceleration is proportional to sinθ.