Year: 2010
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Circular Motion 2
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: 1468.9
Banger Comparisons: 2
$\,$
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The diagram shows
two particles, $P$ and $Q$,
connected by a light inextensible string which passes over a
smooth block fixed to a horizontal table.
The
cross-section of the block is a quarter circle with centre $O$, which
is at the edge of the table, and radius $a$. The angle between
$OP$ and the table is $\theta$.
The masses of $P$ and $Q$ are $m$ and $M $, respectively,
where $m < M$.
Initially, $P$ is held at rest on the table and in contact with the block,
$Q$ is
vertically above $O$, and the string is taut.
Then $P$ is released. Given that, in the subsequent motion,
$P$
remains in contact with the block as $\theta$
increases from $0$ to $\frac12\pi$,
find an expression, in terms of $m$, $M$, $\theta$ and $g$,
for the normal reaction of the block on $P$ and show
that
\[
\frac{m}{M} \ge \frac{\pi-1}3\,.
\]
Less than a fifth of the candidates attempted this, though it was the most popular of the non-Pure questions. Candidates were largely fairly successful or struggled to get started. Some of those failing to get anywhere equated the normal reaction on P to the component of P's weight, completely ignoring the radial acceleration, and others got the sign of the force wrong. Nearly every candidate failed to justify imposing the non-negative condition on the normal reaction when θ = π.