A solid sphere rolls without slipping along the track shown in the figure. The sphere starts from rest from a height h above the bottom of the loop of radius R which is much larger than the radius of the sphere r . The minimum value of h for the sphere to complete the loop is

Step-by-step solution

At the topmost point of the loop minimum value of the linear speed of the centre of the sphere should be : v = gR or translational kinetic energy K T = 1 2 mv 2 = 1 2 mgR . In case of pure rolling of a solid sphere the ratio of rotational to translational kinetic energy is : K R K T = 2 5 ∴ Total kinetic energy at topmost point should be : K = 5 + 2 5 . K T = 7 5 1 2 mgR = 7 1 0 mgR Now from conservation of mechanical energy : 7 1 0 mgR = mg h  - 2 R ∴   h   =   2 . 7   R