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Conical Pendulum proof of constant frequency

  1. Jan 17, 2012 #1
    1. The problem statement, all variables and given/known data
    A pendulum consists of a particle of mass m at the end of a light rigid rod of
    length l, the other end of the rod being freely attached to a stationary point
    0. Let e(t) be a unit vector pointing along the rod, so that the position
    vector relative to 0 of the particle at time t is le(t).
    (a) Show from the principle of linear momentum that

    e x e'' - (g/l) e x k = 0

    where g is the acceleration due to gravity, and k is a unit vector pointing
    vertically upwards. ( e'' represents the second derivative of the unit vector wrt time )

    (b) If e(t) is confined to a vertical plane containing 0, the pendulum is
    called simple. Suppose that

    e(t) = sin θ(t) i - cos θ(t)k,
    where i is a horizontal unit vector. Deduce that

    θ'' + (g/l) sinθ = 0

    (c) If e(t) rotates about k, the pendulum is called conical. Suppose that

    e' = w(t) (k x e(t))

    Deduce that t →w(t) and t →e(t).k are constant throughout the motion.

    If w(0) = w > 0 and e(0).k = -cosθ with 0 < 0 < pi/2, show that

    w = sqrt( g secθ/l)

    Hint. The following identity holds for the triple vector product:

    u x ( v x w) = (u.w)v + (u.v)w

    3. The attempt at a solution

    First two parts are alright, e x e'' + g/l e x k = 0
    so e'' must be = - g/l k
    so le'' = -g k
    a = -gk
    so F = -gm k which is just the force due to gravity, so the original statement is true

    The second part i can do just by working through the differentiation of sinθ i - cosθj, then subbing in and cancelling all the terms.

    the last bit though i think should be
    e' = w( k x e)
    multiply both sides by e
    so e x e' = w( e x (k x e))
    = 1/m ( e x p) = w( e x ( k x e))
    the d/dt of the left is = 0, since the angular momentum is constant,

    then w( (e.e)k -(e.k)e) from the hint must be constant in time
    so d/dt [wk - w(e.k)e ]= 0.

    I can see how this would imply that dw/dt = 0, but not how d(e.k)/dt must also be 0

    Have i done enough and just dont see it, or should i need another proof that (e.k) is constant.


    Thanks in advance for any help you can give.
     
  2. jcsd
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