A plank of mass M rests on a smooth horizontal plane. A sphere of mass m and radius r is placed on the rough upper surface of the plank and the plank is suddenly given a velocity v in the direction of its length. Find the time after which the sphere begins pure rolling, if the coefficient of friction between the plank and the sphere is `mu` and the plank is sufficiently long.

Explanation

To solve the problem of finding the time after which the sphere begins pure rolling on the plank, we can follow these steps:

### Step 1: Analyze the Forces Acting on the Sphere
When the plank is given a velocity \( v \), the sphere experiences a frictional force \( F \) acting in the direction of the plank's motion. This frictional force is responsible for accelerating the sphere. The maximum static friction can be given by:
\[ F = \mu mg \]
where \( \mu \) is the coefficient of friction, \( m \) is the mass of the sphere, and \( g \) is the acceleration due to gravity.
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