Question

A solid sphere is given an angular velocity $\omega$ and kept on a rough fixed incline plane. Then choose the correct statement.

• A. If $\mu =\tan \theta$ then sphere will be in linear equilibrium for some time and after that pure rolling down the plane will start
• B. If $\mu =\tan \theta$ then sphere will move up the plane and frictional force acting all the time will be $2mg$ $sin$.
• C. If $\mu =\frac{\tan \theta }{2}$ there will never be pure rolling (consider inclined plane to be long enough.)
• D. If incline plane is not fixed and it is on smooth horizontal surface then linear momentum of the system (wedge and sphere) can be conserved on horizontal direction.

Answer 1

Answer: (A), (D)

The correct options are
A If $\mu =\tan \theta$ then sphere will be in linear equilibrium for some time and after that pure rolling down the plane will start
D If incline plane is not fixed and it is on smooth horizontal surface then linear momentum of the system (wedge and sphere) can be conserved on horizontal direction.

on inclined plane, the component of gravitational force is $mgsin\theta$, which is countered by the frictional force in the opposite direction, with a maximum limit of $\mu N=\mu mgcos\theta$.

If this limit is sufficient for keeping the sphere in linear equilibrium, it will have angular deceleration, till it starts pure rolling.

So, for linear equilibrium,

$\mu mgcos\theta =mgsin\theta ⟹\mu =tan\theta$.

If the whole system is placed on smooth horizontal surface, so there are no forces on the system in horizontal direction. So the linear momentum in the horizontal direction would be conserved.

Option A and D are correct.