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Pendulum swings into a peg

  1. Apr 14, 2016 #1
    1. The problem statement, all variables and given/known data
    With this problem I have to get the answer: cosθ = r/L * cosα - √(3)/2 * (1 - r/L)
    which in other words mean I need to find angle θ with arccos[r/L * cosα - √(3)/2 * (1 - r/L)].

    Here's the picture:
    11d636180c.jpg



    Lcosθ is the vertical length of the string at its lowest point.

    rcosα is a fraction of that same vertical string in terms of displacement "r" (which is from the start of the string to the peg)

    ∠β is the angle between the peg and the horizon.

    (L-r) sinβ is the height from the end of the peg and the horizontal

    (L-r) cosα is the horizontal length of that same peg.

    (L-r)cosα is the vertical length of the string from the ball to the peg.

    So, this is not really a physics issue but more like a math issue but since this is a physics problem I've decided to put it under here.

    My problem is that I am unable to continue from this point as shown on the picture of my attempt. I don't know where to continue from here on out. I am trying to find "t" for the equation but I am unsure how. Where do I continue from now?

    2. Relevant equations

    Newtonian Position Formula:
    yf = yi +viyt + .5gt2
    xf = xi +vixt + .5gt2

    Energy Equation:
    Work of hand - force of friction * displacement = delta Kinetic Energy + delta Potential Energy

    Wh - fF*d = [.5*mvf2 - .5*mvi2] - [mghf - mghi]

    3. The attempt at a solution
    Picture of Attempt:
    e4bfe92db3.jpg
     
  2. jcsd
  3. Apr 14, 2016 #2

    BvU

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    Hello Stranger, :welcome:

    Your position formula is only valid for uniform acceleration. You don't have that here !
    Can you think of a condition you can impose on ##\beta## ?
     
  4. Apr 14, 2016 #3
    Thanks for the welcome :).
    The only method I can think of when dealing with changing acceleration by breaking it into parts. Each part for every time the value of acceleration changes. I have not yet learned ho to derive very well but I know it exists. As for angle β I am clueless on what to impose.
     
  5. Apr 15, 2016 #4

    BvU

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    Well, then perhaps you can conquer this one without dealing with changing acceleration ?
    The sketch suggests a trajectory, but is it realistic ? Where must the mass run out of sped to fall on the peg ? What would happen if it ran out of speed at e.g. 85 degrees ?
     
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