me (problems with rotating cylinder)

[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.
http://www.geocities.com/andre_pradhana/cylinderkendro2.JPG
When I positioned the cylinder on a flat ramp like the picture below:
http://www.geocities.com/andre_pradhana/cylinderkendro5.JPG
The cylinder will start oscillating back and forth as the weight of the extra mass provide a torque, causing the cylinder to rotate.
http://www.geocities.com/andre_pradhana/cylinderkendro4.JPG
My question is, suppose that the value of $$\Beta$$ 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)?

Suppose that the torque caused by the weight of the smaller cylinder is $$M_1gxsin\theta$$ and the Moment of Inertia is: $$M_2R_2+0.5 M_1r_1^2$$. I can then figured out the equation for angular acceleration, which is: $$\displaystyle{\frac{M_1gxsin\theta}{ M_2R_2+0.5 M_1r_1^2}}$$

However, what I don’t know any formula that relates $$\theta$$ as a function of time. How can I find the time it takes for the smaller cylinder to move from an initial displacement of $$\pi/4$$ befor the cylinder is released until the value of angular displacement is $$\pi$$’ i.e. when it’s directly above the point P, assuming that there is NO friction.

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)? From dimensional analysis, I know that integrating that the integration will give me the value of $$constant\times\omega^2$$

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 there another approach to solve this problem?

Thank you very much for your help...

 

[Astronuc]

First of all, this is a homework problem which belongs in the Homework sections, not in the tutorial section. This thread will be moved.

Secondly, without dispersive forces, e.g. friction, one should obtain a simple harmonic motion, and $\omega$ = $\dot{\theta}$, and $\alpha$ = $\dot{\omega}$.

 

[enigma]

I have moved it to HW help.

kendro, in the future, please don't post threads in multiple forums, and place homework and textbook problems in the homework help section.