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Disk falling while rotating round a point

  1. Aug 22, 2015 #1
    1. The problem statement, all variables and given/known data
    The disk of mass m and radius R is at position (a), upwards. it is free to rotate round point O on it's circumference.
    If falls and reaches state (b), horizontal position.
    What are the velocities and accelerations in position (b)

    2. Relevant equations
    Moment of inertia of a disk: ##I=\frac{1}{2}mR^2##
    Torque of rigid body: ##M=I\alpha##
    Energy of a rigid body: ##E=\frac{1}{2}I\omega^2##
    Centripetal acceleration: ##a_{cen}=\frac{v^2}{R}=\omega^2 R##

    3. The attempt at a solution
    The angular velocity ω is from energy conservation:
    $$mgR=mR^2\left( \frac{1}{2}+1 \right)\omega^2\; \rightarrow \omega^2=\frac{4g}{3R}$$
    COM's velocity is ##v_{cm}=\omega R=\sqrt{\frac{4g}{3R}}R=2\sqrt{\frac{gR}{3}}##
    The radial velocity from the circular motion:
    $$a_{cen}=\omega^2 R=\frac{4g}{3}$$
    Tangential acceleration is from the gravity:
    $$M=I\alpha\rightarrow mgR=\frac{3}{2}mR^2\alpha\rightarrow\alpha=\frac{2g}{3R}$$

    Attached Files:

  2. jcsd
  3. Aug 22, 2015 #2


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    Looks right.
  4. Aug 22, 2015 #3


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    One typo.... where you wrote radial velocity you meant radial acceleration.
  5. Aug 22, 2015 #4
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