Year: 2011
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Circular Motion 2
No solution available for this problem.
The percentages attempting larger numbers of questions were higher this year than formerly. More than 90% attempted at least five questions and there were 30% that didn't attempt at least six questions. About 25% made substantive attempts at more than six questions, of which a very small number indeed were high scoring candidates that had perhaps done extra questions (well) for fun, but mostly these were cases of candidates not being able to complete six good solutions.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
Particles $P$ and $Q$ have masses
$3m$ and $4m$, respectively. They
lie on the outer curved surface of a~smooth
circular cylinder of radius~$a$
which is fixed with its axis horizontal.
They are connected by a light inextensible
string of length $\frac12 \pi a$, which passes over the
surface of the cylinder. The particles and the string all lie
in a vertical plane perpendicular to the axis of the cylinder,
and the axis intersects this plane at $O$.
Initially, the particles are in equilibrium.
Equilibrium is slightly disturbed and $Q$ begins to
move downwards. Show that while the two particles
are still in contact with the cylinder the angle $\theta$
between $OQ$ and
the vertical satisfies
\[
7a\dot\theta^2 +8g \cos\theta + 6 g\sin\theta = 10g\,.
\]
\begin{questionparts}
\item
Given that $Q$ loses contact with
the cylinder first, show that it does so when~$\theta=\beta$,
where $\beta$ satisfies
\[
15\cos\beta +6\sin\beta =10.
\]
\item
Show also that while $P$ and $Q$ are still in contact
with the cylinder
the tension in the string is $\frac {12}7 mg(\sin\theta
+\cos\theta)\,$.
\end{questionparts}
About a sixth of candidates tried this, and on average with slightly less success than question 2. Of the attempts, about a third were close to completely correct, and nearly all the others were barely doing more than grasping at crumbs, reflecting the fact that candidates either did or did not know what they were doing. There was negligible middle ground.