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Pendulum swinging and hitting a peg

  1. Oct 29, 2016 #1
    1. The problem statement, all variables and given/known data
    a pendulum of mass m and length L is pulled back an angle of θ and released. After the pendulum swings through its lowest point it encounters a peg α degrees out and r meters from the top of the string. The mass swings up about the peg until the string becomes slack with the mass falling inward and hitting the peg. Show for this condition cosθ=(r/L)cosα - (sqrt(3)/2)(1-(r/L))
    http://michael-tech.hostzi.com/ec.png [Broken]

    2. Relevant equations
    Energy from start to slack
    mgh = mgh2 + (mv2)/2
    forces on slacking pendulum
    ∑F = mg = (mv2)/L
    3. The attempt at a solution

    so i have gotten the energy from start to where it slacks

    gif.latex?mgh%3Dmgh_%7B2%7D+%5Cfrac%7Bmv%5E2%7D%7B2%7D.gif

    5CTheta%20%3Dg%28L-rcos%5Calpha%20+%20%28L-r%29sin%5Cbeta%29+%5Cfrac%7Bv%5E2%7D%7B2%7D.gif

    with forces when it slacks i get
    gif.latex?gsin%5Cbeta%20%3D%5Cfrac%7Bv%5E2%7D%7BL-r%7D.gif solving for v^2 gif.latex?v%5E2%3Dg%28L-r%29sin%5Cbeta.gif

    plugging and chugging with energy i get
    gif.latex?sin%5Cbeta%3D%5Cfrac%7B2%28Lcos%5Ctheta-rcos%5Calpha+L%29%7D%7B3%28L-r%29%7D.gif if image is broken http://michael-tech.hostzi.com/e1.gif [Broken]

    plugging back in to forces i get

    gif.latex?%5Cfrac%7B2g%7D%7B3%7D%28Lcos%5Ctheta-rcos%5Calpha+L%29%3Dv%5E2.gif

    now it is just a projectile and this is where i am stuck. Here is what i know at this point.
    gif.latex?x_%7B0%7D.gif = 0 gif.latex?y_%7B0%7D.gif = 0
    gif.latex?x%3D%28L-r%29cos%5Cbeta.gif gif.latex?y%3D%28L-r%29sin%5Cbeta.gif
    gif.latex?v_%7B0x%7D%20%3D%20%3F%20v%20cos%5Cbeta%3F.gif gif.latex?v_%7B0y%7D%20%3D%20%3F%20v%20sin%5Cbeta%3F.gif
    gif.latex?v_%7Bx%7D.gif = dont think i need, think its 0 though vy = dont think i need, think its 0 though
    a = 0 a = -g
    t=t

    I am getting stuck at the Vnot should it be vcosβ. Also does the above make sense so far
     
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Oct 29, 2016 #2

    haruspex

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    The initial height is not L cos(θ), unless you are measuring height downwards.
    You do need to care about both velocity components at the point where the string goes slack. What you not care about is final velocities, so use the vertical SUVAT equation that does not involve final velocity. Time only matters to the extent that x and y must coincide with the peg at the same point in time.
     
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