Year: 2010
Paper: 3
Question Number: 10
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
About 80% of candidates attempted at least five questions, and well less than 20% made genuine attempts at more than six. Those attempting more than six questions fell into three camps which were those weak candidates who made very little progress on any question, those with four or five fair solutions casting about for a sixth, and those strong candidates that either attempted 7th or even 8th questions as an "insurance policy" against a solution that seemed strong but wasn't, or else for entertainment!
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A small bead $B$, of mass $m$,
slides without friction on a fixed horizontal ring of
radius $a$. The centre of the ring is at $O$. The bead is attached by a light
elastic string to a fixed
point $P$ in the plane of the ring such that $OP = b$, where $b > a$.
The natural length of the elastic string is $c$, where $c < b - a$, and its
modulus of elasticity is $\lambda$.
Show that the equation of motion of the bead is
\[
ma\ddot \phi =
-\lambda\left( \frac{a\sin\phi}{c\sin\theta}-1\right)\sin(\theta+\phi)
\,,
\]
where
$\theta=\angle BPO$ and $\phi=\angle BOP$.
Given that $\theta$ and $\phi$ are small, show that
$a(\theta+\phi)\approx
b\theta$. Hence find the period of
small oscillations about the
equilibrium position $\theta=\phi =0$.
Only 5% of the candidates attempted this and it was another case of nearly all or nothing. Even the mostly successful candidates rarely handled the small oscillation algebraic manipulation correctly, often overlooking using the small angle result throughout the expression, so very few obtained the correct period though the principle was understood.