A solid sphere , a hollow sphere and a disc , all having same mass and radius , are placed at the top of an incline and released . The friction coefficient between the objects and the incline are same and not sufficient to allow pure rolling . Prove that time taken in reaching the bottom is same for all .

A solid sphere , a hollow sphere and a disc , all having same mass and

1.	A solid sphere , a hollow sphere and a disc , all having same mass and radius , are placed at the top of an incline and released . The friction coefficient between the objects and the incline are same and not sufficient to allow pure rolling . Prove that time taken in reaching the bottom is same for all .
Answer» Solution :As the body does not undergo pure rolling , it means friction acting is kinetic friction = `mu N` . Normal to the inclined plane , the body is in equilibrium , hence N = `Mg COS theta` Along the inclined plane , `Mg sin theta - mu N = Ma implies Ma = Mg sin theta - mu Mg cos theta` or `a = G sin theta - mg cos theta` As MOMENT of INERTIA , does not feature in the acceleration equation , all bodies whether they are SOLID sphere , hollow sphere or disc will have the same acceleration . they will take the same time to reach the bottom .