University of Cambridge Mathematics Spherical Pendulum Question

Question Description

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

1 attachmentsSlide 1 of 1attachment_1attachment_1

Unformatted Attachment Preview

Ꮖ
(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

Purchase answer to see full
attachment

Explanation & Answer:
1 Answer

Tags:
algebra

mathematics

Spherical Pendulum

Student has agreed that all tutoring, explanations, and answers provided by the tutor will be used to help in the learning process and in accordance with Studypool’s honor code & terms of service.