Year: 2007
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Work, energy and Power 2
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
Two small beads, $A$ and $B$, each of mass $m$, are threaded
on a smooth horizontal circular hoop of radius $a$ and centre $O$.
The angle $\theta$ is the acute angle determined by
$2\theta = \angle AOB$.
The beads are connected by a light straight spring.
The energy
stored in the spring
is
\[
mk^2 a^2(\theta - \alpha)^2,
\]
where
$k$ and $\alpha$ are constants satisfying
$k>0$ and $\frac \pi 4< \alpha<\frac\pi2$.
The spring is held in compression with $\theta =\beta$
and then
released.
Find the period of oscillations in the two cases that arise according
to the value of $\beta$ and state the value of $\beta$ for which
oscillations do not occur.
This was little attempted. Some did struggle through to the solution of the differential equation, but the appreciation of the three possible cases eluded them.