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Thread: [SOLVED] Difficult rotational mechanics question?

  1. #1
    Super Member fardeen_gen's Avatar
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    [SOLVED] Difficult rotational mechanics question?

    A uniform circular cylinder of mass $\displaystyle m$ and radius $\displaystyle r$ is given an initial angular velocity $\displaystyle \omega_{0}$ and no initial translational velocity. It is placed in contact with a plane inclined at an angle $\displaystyle \alpha$ to the horizontal. If there is a coefficient of friction $\displaystyle \mu$ for sliding between the cylinder and plane, find the distance the cylinder moves up before sliding stops. Also, calculate the maximum distance it travels up the plane. Assume $\displaystyle \mu > \tan\alpha$

    Answer:
    Spoiler:
    $\displaystyle d_{1} = \frac{r^2\omega_{0}^2(\mu \cos \alpha - \sin \alpha)}{2g(3\mu\cos \alpha - \sin \alpha)^2}$
    $\displaystyle d_{max} = \frac{r^2\omega_{0}^2(\mu \cos \alpha - \sin \alpha)}{4g\sin \alpha(3\mu\cos \alpha - \sin \alpha)}$


    Please help!
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  2. #2
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    This is actually quite simple to solve and here is how you do it:

    Spoiler:

    To begin with here's a diagram of the initial state of the cylinder:


    For the duration of time when there is still sliding taking place then the friction force is at maximum and therefore the acceleration is a constant and so is the torque on the cylinder. This means the following equations of motion hold:

    $\displaystyle \omega = \omega_0 + \ddot{\theta}t $ and $\displaystyle v = a t $

    where $\displaystyle v$ and $\displaystyle \omega$ are the velocity and angular velocity at the point when sliding stops respectively. Also
    $\displaystyle v = r \omega$.
    Hence, $\displaystyle v$ can be found in terms of $\displaystyle \ddot{\theta}$ and $\displaystyle a$ as
    $\displaystyle v = \frac{a}{a-r \ddot{\theta}} r \omega_0$ .

    This equation for the velocity can then be used with the equation of motion relating velocity to distance to find $\displaystyle d_1$, i.e.
    $\displaystyle v^2 = u^2 +2aS$ so then
    $\displaystyle \color[rgb]{0,0,1}d_1 = \frac{a}{(a-r\ddot{\theta})^2} \frac{r^2 \omega_{0}^{2}}{2}$.

    Now all you need to do is find $\displaystyle \ddot{\theta}$ and $\displaystyle a$ and substitute in the above equation. These two can now be found by resolving forces in the diagram.
    Friction force is given by

    $\displaystyle F_\mu = \mu R$

    Resolving perpendicular to the plane we have:

    $\displaystyle R = m g \cos \alpha$.

    Resolving along the plane

    $\displaystyle F_\mu - mg \sin \alpha = m a$ .

    From these three equations we find that

    $\displaystyle \color[rgb]{0,0,1} a = (\mu \cos \alpha - \sin \alpha) g$.

    Finally the torque is given by:

    $\displaystyle I \ddot{\theta} = -r F_\mu$ where $\displaystyle \color[rgb]{0,0,1}I = \frac{1}{2} m r^2$.

    Hence

    $\displaystyle \color[rgb]{0,0,1} \ddot{\theta} = - \frac{2 \mu g \cos \alpha}{r}$ .

    So that gives you $\displaystyle d_1$ easily enough!


    For the second part you can just use the conservation of energy to work out the further distance travelled, $\displaystyle d_2$, before stopping. So equating kinetic energy lost to the potential energy gained gives:
    $\displaystyle \color[rgb]{0,0,1}mg \, d_2 \, \sin \alpha = \frac{1}{2} I \omega^2 +\frac{1}{2} m v^2$

    then

    $\displaystyle d_{max} = d_1 +d_2$ .
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