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.(adsbygoogle = window.adsbygoogle || []).push({});

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 [tex]\Beta[/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)?

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

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

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

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 there another approach to solve this problem?

Thank you very much for your help....

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# Homework Help: Please Help me (problems with rotating cylinder)

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