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Velocity & Accel of a mass inside a slot on a rolling disk

  1. Oct 7, 2015 #1
    1. The problem statement, all variables and given/known data
    The disk in Figure 3.30 of radius R rolls without slipping with constant angular velocity Ω. Carved inside the disk is a slot and a mass moves inside the slot. Denoting the position of the mass inside the slot by s, calculate the velocity and acceleration of the mass as a function of θ.

    ZyxGcmB.png


    2. Relevant equations

    v_A = RΩ
    v_B = v_A + v_B/A
    a = dv/dt


    3. The attempt at a solution

    v_A = R*Ω, which is fairly obvious since the disk does not slip. v_B is then = R*Ω + v_B/A
    Finding v_B/A is where I'm having issues. My first thought is that it would be Ω*√((5R/8)^2+s^2), but this isn't in terms of θ and is only angular velocity. I've thought about trying to relate θ using the relation tan(θ) = 8s/(5R), but I don't know how to fit this into the equation.

    I haven't taken a look at finding the acceleration yet, but I imagine I could just take the time derivative of the velocity equation to get the acceleration.
     
  2. jcsd
  3. Oct 8, 2015 #2

    haruspex

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    It might help to think in terms of instantaneous centre of rotation. Where is that for a rolling disc?
     
  4. Oct 8, 2015 #3
    The instantaneous center is at the point where it contacts the ground, or in this case, point C. I don't know how I would relate this to the velocity of the mass since the mass's instant center is changing as the disk rolls.
     
  5. Oct 9, 2015 #4

    haruspex

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    Sure, there is a velocity relative to the disk as well, but the direction of that is constrained. So it remains to find the total speed. How might you do that?
     
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