A circular cylinder of radius $a$, length $L$ and mass $m$ is rolling along a surface. Show that its moment of inertia is given by $\frac{1}{2} m a^{2}$.

At $t=0$ the cylinder is at the bottom of a slope making an angle $\alpha$ to the horizontal, and is rolling with velocity $V$ and angular velocity $V / a$. Assuming slippage does not occur, determine the position of the cylinder as a function of time. What is the maximum height that the cylinder reaches?

The frictional force between the cylinder and surface is given by $\mu m g \cos \alpha$, where $\mu$ is the friction coefficient. Show that the cylinder begins to slip rather than roll if $\tan \alpha>3 \mu$. Determine as a function of time the location, speed and angular velocity of the cylinder on the slope if this condition is satisfied. Show that slipping continues as the cylinder ascends and descends the slope. Find also the maximum height the cylinder reaches, and its speed and angular velocity when it returns to the bottom of the slope.