# Rolling Motion

#### kendro

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 $$\alpha\times\theta$$:
$$\omega_t^2=\omega_0^2+2\alpha\theta$$

Does it mean that if I integrate:
$$\int_ {\pi/4}^{\pi} \alpha d\theta$$
Will I get the value of $$0.5\times\omega_t^2$$ when $$\theta$$ is $$\pi$$? (with the assumption that the value of $$\omega_0$$ 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?

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#### vaishakh

But that will not help you to get the actual time taken to reach the angular displacement Pi. Your formula us correct only if angular acceleratin is a constant I would think.
Your method of approach is perfect. You find the torque each time. The distance of the object from centre of mass is R2. The force acting on that point after the block has undergone some amount of angular displacement theta, The force acting on the wheel is Wcos(theta) where W is the weight o the block. Wsin(thata) is the force towards the centre. Now integrate to get the proper time.

"Rolling Motion"

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