# Rolling Motion

Hi. I have a problem about rolling motion. Suppose that I have a large hollow cylinder. A smaller solid cylinder is embedded inside the larger hollow cylinder.
When I positioned the cylinder on a flat ramp like the picture below:
The cylinder will start oscillating back and forth as the weight of the extra mass provide a torque, causing the cylinder to rotate. My question is, suppose that the value of $$\theta$$ initially was $$\pi/4$$ before the cylinder is released and start rolling, how can I calculate the time it takes before the $$\theta$$ reaches a value of $$\pi$$ (when the extra mass is directly above the point P)?

If I figured out the function of angular acceleration in terms of angular displacement, what will I get when I integrate the function of angular acceleration in respect to angular displacement, since the value of angular acceleration is always changing as $$\theta$$ changes.

I know that there’s a formula relating $$\alpha\times\theta$$:
$$\psi_t^2=\psi_0^2+2\alpha\theta$$

Does it mean that if I integrate:
$$\int_ {\pi/4}^{\pi} \alpha d\theta$$
Will I get the value of $$0.5\times\psi_t^2$$ when $$\theta$$ is $$\pi$$? (with the assumption that the value of $$\psi_0$$ initially is 0 rad/s)

If that’s true, then I can figured out the average angular acceleration to calculate the time it takes for the extra mass to travel from $$\pi/4$$ to $$\pi$$. Is that right?