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Please Help me (problems with rotating cylinder)

  1. Feb 11, 2006 #1
    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.
    [​IMG]
    When I positioned the cylinder on a flat ramp like the picture below:
    [​IMG]
    The cylinder will start oscillating back and forth as the weight of the extra mass provide a torque, causing the cylinder to rotate.
    [​IMG]
    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....
     
  2. jcsd
  3. Feb 11, 2006 #2

    Astronuc

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    Staff: Mentor

    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 [itex]\omega[/itex] = [itex]\dot{\theta}[/itex], and [itex]\alpha[/itex] = [itex]\dot{\omega}[/itex].
     
  4. Feb 17, 2006 #3

    enigma

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    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
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