A cylinder is rotating with angular velocity `omega_(0)` and is gently put on a rough horizontal floor. Assume mass of the cylinder is m and radius R. Calculate the velocity of cylinder when it starts pure rolling on the surface.

Explanation

To solve the problem of a cylinder rotating with an initial angular velocity \(\omega_0\) and transitioning to pure rolling on a rough horizontal surface, we can follow these steps:

### Step 1: Understand the Initial Conditions
The cylinder is initially rotating with an angular velocity \(\omega_0\) and has a radius \(R\). When it is placed on the rough surface, the bottommost point of the cylinder tends to move backward due to its rotation.

### Step 2: Identify the Forces Acting on the Cylinder
When the cylinder touches the rough surface, a frictional force \(f\) acts on it in the forward direction. This frictional force is responsible for the acceleration of the center of mass of the cylinder.

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