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What have I done wrong? (torque and angular momentum)

  1. Mar 9, 2014 #1
    [SOLVED] What have I done wrong? (torque and angular momentum)

    1. The problem statement, all variables and given/known data
    ZjybySY.gif
    A billiardball is hit from rest by the cue at height "h" from the table with a force F in the time interval Δt. The mass M and radius R of the ball is known as well as the moment of inertia which is [itex]I=\frac{2}{5}MR^2[/itex]
    Find an expression for the height "h" at which the billiardball will roll without slipping when it is hit.


    2. Relevant equations
    Condition for roll with no slipping:
    [itex]v_{CM}=R\omega[/itex]


    3. The attempt at a solution
    I start by finding an expression for [itex]v_{cm}[/itex].

    Center of mass differentiated by Δt gives:
    [itex]v_{CM}=\frac{Mv}{M}=v[/itex]


    Newton's 2nd law:
    [itex]F=M\frac{v}{Δt}[/itex]

    Isolating velocity gives:
    [itex]v=MFΔt[/itex]

    Since the velocity is equal to the velocity of the center of mass:
    [itex]v_{CM}=MFΔt[/itex]

    Now I find an expression for the angular velocity.

    The net torque is given by:
    [itex]∑τ=Iα[/itex]

    The only force is the force F from the cue which gives the torque [itex]τ=F(h-R)[/itex] where (h-R) is the perpendicular length from the force F to the center of mass of the ball.
    [itex]F(h-R)=I\frac{\omega}{Δt}[/itex]

    The angular velocity is:
    [itex]\omega=\frac{F(h-R)Δt}{I}[/itex]

    Now I insert the velocity of CM and the angular velocity into the rolling without slip equation:
    [itex]MFΔt=R\frac{F(h-R)Δt}{I}[/itex]

    And I end up with:
    [itex]h=R(2/5M^2+1)[/itex]

    But my expression for the height has the mass squared in it. What did I do wrong?
     
    Last edited: Mar 9, 2014
  2. jcsd
  3. Mar 9, 2014 #2

    haruspex

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    Try that step again.
    (Do you know how to do dimensional analysis? That's a very useful way to sanity-check an equation.)
     
  4. Mar 9, 2014 #3
    That's it! Now the answer is correct, thanks! :)
     
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