# Rolling Down Rolling Down Rolling Down

**Classical Mechanics**Level 5

A Spinning cylinder of mass \(m=5 \text{kg} \) and radius \(R\), is lowered with the angular velocity \(\omega_0\) in the clockwise direction on a rough inclined plane of angle \(30^{ \circ } \) with the horizontal and coefficient of Friction \(\mu\).

The cylinder is released at a height of \(3R\) from the Horizontal.

Evaluate the total time taken by the cylinder to reach the bottom of the incline (to 3 decimal places).

**Details and Assumptions**:

\(g = 10 \text{m}/{s}^{2} \) , \(\mu = \dfrac{1}{\sqrt{3}}\) , \(R = 5 \text{m} \) , \(\omega_0 = 2 \text{rad}/s \)