Conical Pendulum proof of constant frequency

[tpb77]

Homework Statement

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

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 don't see it, or should i need another proof that (e.k) is constant.Thanks in advance for any help you can give.

 

[mark726]

Your attempt at solving this problem is mostly correct. However, there are a few things that need to be clarified.

Firstly, when you say "the second part i can do just by working through the differentiation of sinθ i - cosθj", it is important to note that the differentiation should be done with respect to time, since we are dealing with a dynamic system. So the correct statement would be "the second part I can do by taking the second derivative of sinθ i - cosθk with respect to time".

Secondly, in the last part, you have correctly shown that the angular momentum is constant (since the derivative of the triple vector product is zero). However, you cannot directly conclude that the derivative of w with respect to time is zero. This is because the derivative of w with respect to time would be equal to the derivative of the triple vector product with respect to time, which is not necessarily zero. What you can conclude is that the derivative of w with respect to time is equal to the derivative of the triple vector product with respect to time divided by m, which is a constant. This means that dw/dt is constant, but not necessarily zero.

To show that d(e.k)/dt is also constant, you can use the fact that (e.k) is a constant (since e and k are both unit vectors and their dot product is equal to cosθ), and therefore its derivative with respect to time is equal to zero. This means that d(e.k)/dt is also a constant, and since it is equal to zero at t=0, it must remain zero for all values of t.

I hope this clarifies your doubts. Please let me know if you need further assistance.