University of Cambridge Mathematics Spherical Pendulum Question

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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
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
a cos 0

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Spherical Pendulum

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