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Math Help - Rotational Dynamics

  1. #1
    Super Member fardeen_gen's Avatar
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    Rotational Dynamics

    One-fourth length of a uniform rod of mass m and length l is placed on a rough horizontal surface and it is held stationary in horizontal position by means of a light thread. The thread is then burnt and the rod starts rotating about the edge. Find the angle between the rod and the horizontal when it is about to slide on the edge. The coefficient of friction between the rod and the surface is \mu.

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    Last edited by fardeen_gen; May 4th 2009 at 08:30 PM.
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  2. #2
    Super Member fardeen_gen's Avatar
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    Done!

    Using Parallel-axis theorem, Moment of inertia of the rod about pivot point:
    I_{P} = \frac{ml^2}{12} + m\left(\frac{l}{4}\right)^2 = \frac{7ml^2}{48}
    ( I_{CM} = \frac{ml^2}{12})

    Imagine the rod is tilted at an angle \theta w.r.t the horizontal. If you draw an F.B.D, there are three forces on the rod: mg(downward and acting on CM), F_{\perp}(reaction force - \perp to the rod and acting on the pivot point outward), and F_{fric}( \parallel to the rod and pointing to the left, but also upward at an angle \theta).

    Considering the torque due to gravity about pivot point and hence finding angular acceleration as a function of \theta,

    \frac{1}{4}mgl\cos \theta = I_{P}\alpha(\theta)\ \Rightarrow \alpha(\theta) = \frac{12g}{7l}\cos \theta

    Considering torque due to reaction force about CM,


    F_{\perp}\cdot \frac{l}{4} = I_{CM}\alpha(\theta)\\ \Rightarrow F_{\perp} = \frac{4}{7}mg\cos \theta


    F_{fric} = \mu F_{\perp} = \frac{4}{7}\mu mg\cos \theta


    Using conservation of energy principle and hence finding angular velocity as a function of \theta,

    mg\frac{l}{4}\sin \theta = \frac{1}{2}I_{P}\omega(\theta)^2\\ \Rightarrow \omega(\theta)^2 = \frac{24g}{7l}\sin \theta


    Considering the centripetal force that F_{fric} provides,

    F_{fric} - mg\sin \theta = m\omega(\theta)^2\frac{l}{4}
    \Rightarrow \frac{4}{7}\mu mg\cos \theta - mg\sin \theta = \frac{6}{7}mg\sin \theta
    \Rightarrow \boxed{\theta = \arctan\left(\frac{4\mu}{13}\right)}
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