Year: 2003
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Simple Harmonic Motion
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1503.8
Banger Comparisons: 2
A particle $P$ of mass $m$ is constrained to move on a vertical
circle of smooth wire with centre~$O$ and of radius $a$.
$L$ is the lowest point of the circle and $H$ the highest and
$\angle LOP = \theta\,$. The particle is attached to $H$ by an
elastic string of natural length $a$ and modulus of elasticity~$\alpha mg\,$,
where $\alpha > 1\,$. Show that, if $\alpha>2\,$, there is an
equilibrium position with $0<\theta<\pi\,$.
Given that $\alpha =2+\sqrt 2\,$, and that
$\displaystyle \theta = \tfrac{1}{2}\pi + \phi\,$, show that
\[
\ddot{\phi} \approx -\frac{g (\sqrt2+1)}{2a }\, \phi
\]
when $\phi$ is small.
For this value of $\alpha$,
explain briefly what happens to the particle if it
is given a small displacement when $ \theta = \frac{1}{2}\pi$.