Year: 2008
Paper: 2
Question Number: 11
Course: LFM Pure and Mechanics
Section: Friction
No solution available for this problem.
There were around 850 candidates for this paper – a slight increase on the 800 of the past two years – and the scripts received covered the full range of marks (and beyond!). The questions on this paper in recent years have been designed to be a little more accessible to all top A-level students, and this has been reflected in the numbers of candidates making good attempts at more than just a couple of questions, in the numbers making decent stabs at the six questions required by the rubric, and in the total scores achieved by candidates. Most candidates made attempts at five or more questions, and most genuinely able mathematicians would have found the experience a positive one in some measure at least. With this greater emphasis on accessibility, it is more important than ever that candidates produce really strong, essentially-complete efforts to at least four questions. Around half marks are required in order to be competing for a grade 2, and around 70 for a grade 1. The range of abilities on show was still quite wide. Just over 100 candidates failed to score a total mark of at least 30, with a further 100 failing to reach a total of 40. At the other end of the scale, more than 70 candidates scored a mark in excess of 100, and there were several who produced completely (or nearly so) successful attempts at more than six questions; if more than six questions had been permitted to contribute towards their paper totals, they would have comfortably exceeded the maximum mark of 120. While on the issue of the "best-six question-scores count" rubric, almost a third of candidates produced efforts at more than six questions, and this is generally a policy not to be encouraged. In most such cases, the seventh, eighth, or even ninth, question-efforts were very low scoring and little more than a waste of time for the candidates concerned. Having said that, it was clear that, in many of these cases, these partial attempts represented an abandonment of a question after a brief start, with the candidates presumably having decided that they were unlikely to make much successful further progress on it, and this is a much better employment of resources. As in recent years, most candidates' contributing question-scores came exclusively from attempts at the pure maths questions in Section A. Attempts at the mechanics and statistics questions were very much more of a rarity, although more (and better) attempts were seen at these than in other recent papers.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A wedge of mass $km$ has the shape (in cross-section) of a right-angled
triangle. It stands on a smooth horizontal surface with one face
vertical. The inclined face makes an angle $\theta$ with the
horizontal surface.
A particle $P$, of mass $m$, is placed on
the inclined face and released from rest.
The horizontal face of the wedge
is smooth, but the inclined face is rough
and the coefficient of friction between $P$ and
this face is $\mu$.
\begin{questionparts}
\item When
$P$ is released, it slides down the inclined
plane at an acceleration $a$ relative to the wedge. Show that
the acceleration of the wedge is
\[
\frac {a \cos\theta}{k+1}\,.
\]
To a
stationary observer, $P$ appears to descend along a straight line
inclined at an angle~$45^\circ$ to the horizontal. Show that
\[
\tan\theta = \frac k {k+1}\,.
\]
In the case $k=3$, find an expression for $a$ in terms of $g$ and $\mu$.
\item What happens when $P$ is released if $\tan\theta \le \mu$?
\end{questionparts}
This question attracted under 100 attempts and a mean mark of under 3. The strong complaint I have made in the Report over recent years has consistently been that candidates' efforts on such questions have been seriously compromised by a disturbing inability to draw a decent diagram at the outset. I'm afraid that this was a major stumbling-block to successful progress with this question this year also. It was also a bit of a problem that candidates tended to confuse the acceleration of P relative to the wedge with its absolute acceleration relative to the stationary surface on which the wedge stood (say). As few decent attempts were made, it is difficult to be very specific about what went on otherwise.