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I’m working on a mathematics multi-part question and need an explanation and answer to help me learn.

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(The spherical pendulum) A particle of mass m > 0 is suspended by a light thread

of length a from the origin (but is not required to swing in a fixed plane). Its

position is described by

1 = a sin cos ve, +a sin sin vez – acos Dez.

where 0 is the angle between I and the downward vertical.

(a) Show that the kinetic energy of the particle may be expressed in terms of the

generalised coordinates 0, and their derivatives by

1

T= maº (0 + vº sinº 0)

(b) If the only external force is the gravitational force F = -mgez (with g > 0)

obtain the two Lagrange equations for the system. Show that one of them

entails that the quantity l = o sinO is constant, and that the other may be

rewritten as

Ö – 2 sin cos 0 = -9 sin

(c) If the motion of the pendulum bob is such that its height is unchanging (i.e.

Ö=0), show that the angular velocity w = 4 is

w=

9

a cos 0

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Tags:

algebra

mathematics

Spherical Pendulum

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