Year: 2008
Paper: 1
Question Number: 11
Course: LFM Pure and Mechanics
Section: Friction
No solution available for this problem.
There were significantly more candidates attempting this paper this year (an increase of nearly 25%), but many found it to be very difficult and only achieved low scores. The mean score was significantly lower than last year, although a similar number of candidates achieved very high marks. This may be, in part, due to the phenomenon of students in the Lower Sixth (Year 12) being entered for the examination before attempting papers II and III in the Upper Sixth. This is a questionable practice, as while students have enough technical knowledge to answer the STEP I questions at this stage, they often still lack the mathematical maturity to be able to apply their knowledge to these challenging problems. Again, a key difficulty experienced by most candidates was a lack of the algebraic skill required by the questions. At this level, the fluent, confident and correct handling of mathematical symbols (and numbers) is necessary and is expected; many students were simply unable to progress on some questions because they did not know how to handle the algebra. There were of course some excellent scripts, full of logical clarity and perceptive insight. It was also pleasing that one of the applied questions, question 13, attracted a very large number of attempts. However, the examiners were again left with the overall feeling that some candidates had not prepared themselves well for the examination. The use of past papers and other available resources to ensure adequate preparation is strongly recommended. A student's first exposure to STEP questions can be a daunting, demanding experience; it is a shame if that takes place during a public examination on which so much rides.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A straight uniform rod has mass $m$. Its ends $P_1$ and
$P_2$ are attached to small light rings that are constrained to move
on a rough rigid circular wire with centre $O$
fixed in a vertical plane, and the angle
$P_1OP_2$ is a right angle.
The rod
rests with $P_1$ lower than $P_2$, and with both ends lower than $O$.
The coefficient
of friction between each of the rings and the wire is $\mu$.
Given that the rod is in limiting equilibrium (i.e. \ on the point of
slipping at both ends),
show that
\[
\tan \alpha = \frac{1-2\mu -\mu^2}{1+2\mu -\mu^2}\,,
\]
where $\alpha$ is the angle between $P_1O$ and the vertical
($0<\alpha<45^\circ$).
Let $\theta$ be
the acute angle between the rod and the horizontal.
Show that $\theta =2\lambda$, where
$\lambda $ is defined by
$\tan \lambda= \mu$ and $0<\lambda<22.5^\circ$.
This was the least popular of the mechanics questions, and of the candidates who attempted it, only a handful made any progress beyond drawing a usually incorrect sketch and writing down some equations. The majority were aware that F = µR as the equilibrium is limiting. Unfortunately, though, they often either missed forces from their diagram or drew at least one of the frictional forces in the wrong direction. Another frequent problem was that they labelled both normal reaction forces with the same variable R, thereby implicitly implying that they are equal, whereas this is not the case. A small annoyance was the number of candidates who used the notation Fr for friction; an unhealthy practice as it can so easily be confused with F × r. Also, several failed to mark the angle α correctly on their diagram. After this, a small number of candidates correctly resolved in two directions and took moments. Those who understood how to then manipulate the resulting equations to eliminate most of the variables went on to produce essentially perfect solutions, whereas everyone else became stuck at this point and found themselves unable to progress any further. No attempts using the theorem regarding three forces on a large body were seen, which is a shame, as it made the problem significantly easier.