A sphere of mass m attached to a spring on incline plane as shown in figure is held in unstretched position of spring. Suddenly sphere is left free what is the maximum extension of spring if friction allows only rolling of sphere about horizontal diametre ?

Explanation

At the point of maximum extension sphere is at rest. No work is done by frictional force in rolling
`rArr` Loss in gravitational PE =Gain in PE of spring
`rArr (1)/(2)kx^(2)=mhxsin theta rArr x=(2mg sin theta)/(k)`