Solving Rolling Motion of Embedded Cylinder on Ramp

[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/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]\psi_t^2=\psi_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\psi_t^2[/tex] when [tex]\theta[/tex] is [tex]\pi[/tex]? (with the assumption that the value of [tex]\psi_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?

 

[lippyka]

Otherwise, can anyone tell me the best way to calculate the time it takes for the extra mass to travel from \pi/4 to \pi?

 

[quicknote]

I can provide some insight on solving this problem. First, it is important to understand the concept of rolling motion. Rolling motion is a combination of rotation and translation, where an object moves without slipping. In this case, the large hollow cylinder is rotating and translating down the ramp while the smaller solid cylinder is rotating within it.

To calculate the time it takes for the \theta to reach a value of \pi, we need to consider the forces acting on the system. The main force is gravity, which is causing the cylinder to roll down the ramp. The mass of the embedded cylinder also contributes to the torque and affects the motion.

To start, we can use the torque equation, \tau=I\alpha, where \tau is the torque, I is the moment of inertia, and \alpha is the angular acceleration. We can also use the equation for rotational kinetic energy, K=\frac{1}{2}I\omega^2, where K is the kinetic energy, I is the moment of inertia, and \omega is the angular velocity.

Using these equations, we can derive the equation for angular acceleration in terms of angular displacement, \theta. This equation will involve the mass of the embedded cylinder, the mass of the large hollow cylinder, and the radius of the cylinders.

Next, we can integrate this equation from the initial \theta value of \pi/4 to the final value of \pi to get the average angular acceleration. This average acceleration can then be used to calculate the time it takes for the cylinder to reach \pi radians.

In summary, to calculate the time it takes for the embedded cylinder to reach \pi radians, we need to consider the forces and use equations for torque and rotational kinetic energy. By integrating the equation for angular acceleration and using the average value, we can determine the time it takes for the cylinder to reach \pi radians.