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/cylinderkendro3.JPG

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 [tex]\theta[/tex] initially was [tex]\pi/4[/tex] before the cylinder is released and start rolling, how can I calculate the time it takes before the [tex]\theta[/tex] reaches a value of [tex]\pi[/tex] (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 [tex]\theta[/tex] changes.

I know that there’s a formula relating [tex]\alpha\times\theta[/tex]:

[tex]\omega_t^2=\omega_0^2+2\alpha\theta[/tex]

Does it mean that if I integrate:

[tex]\int_ {\pi/4}^{\pi} \alpha d\theta[/tex]

Will I get the value of [tex]0.5\times\omega_t^2[/tex] when [tex]\theta[/tex] is [tex]\pi[/tex]? (with the assumption that the value of [tex]\omega_0[/tex] 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 [tex]\pi/4[/tex] to [tex]\pi[/tex]. Is that right?