Classical Mechanics (Kibble & Berkshire) · Problem 3.9

Problems in particle dynamics — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · gravitation-continuous-distributions

Problem

A uniform wire of mass

M

has the form of a circle of radius

a

and a particle of mass

m

lies in the plane of the wire at a distance

b

(

b < a

) from the centre

O

. Show that the gravitational force exerted by the wire on the particle (in the direction

OP

) is given by

F = \frac{mMG}{2\pi a^2} \int_0^{2\pi} \frac{(\cos\theta - \xi)\,d\theta}{\{1 + \xi^2 - 2\xi\cos\theta\}^{3/2}},

where the dimensionless distance

\xi = b/a

. Use computer assistance to plot the graph of (dimensionless)

F

against

\xi

for

0 \leq \xi \leq 0.8

and confirm that

F

is positive for

\xi > 0

. Is the position of equilibrium at the centre of the circle stable? Could the rings of Saturn be solid?