Classical Mechanics (Kibble & Berkshire) · Problem 6.20

Orbits in a central field — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · constrained-motion-string

Problem

A smooth horizontal table has a vertical post fixed to it which has the form of a circular cylinder of radius

a

. A light inextensible string is wound around the base of the post (so that it does not slip) and its free end of the string is attached to a particle that can slide on the table. Initially the unwound part of the string is taut and of length

4a/3

. The particle is then projected horizontally at right angles to the string so that the string winds itself on to the post. How long does it take for the particle to hit the post? [You may make use of the formula

\int(1+\phi^2)^{1/2}d\phi = \tfrac{1}{2}\phi(1+\phi^2)^{1/2} + \tfrac{1}{2}\sinh^{-1}\phi.\,]

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