Year: 2011
Paper: 2
Question Number: 8
Course: UFM Mechanics
Section: Centre of Mass 2
No solution available for this problem.
There were just under 1000 entries for paper II this year, almost exactly the same number as last year. After the relatively easy time candidates experienced on last year's paper, this year's questions had been toughened up significantly, with particular attention made to ensure that candidates had to be prepared to invest more thought at the start of each question – last year saw far too many attempts from the weaker brethren at little more than the first part of up to ten questions, when the idea is that they should devote 25-40 minutes on four to six complete questions in order to present work of a substantial nature. It was also the intention to toughen up the final "quarter" of questions, so that a complete, or nearly-complete, conclusion to any question represented a significant (and, hopefully, satisfying) mathematical achievement. Although such matters are always best assessed with the benefit of hindsight, our efforts in these areas seem to have proved entirely successful, with the vast majority of candidates concentrating their efforts on four to six questions, as planned. Moreover, marks really did have to be earned: only around 20 candidates managed to gain or exceed a score of 100, and only a third of the entry managed to hit the half-way mark of 60. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were attempted by almost all candidates; 3 and 4 by around three-quarters of them; 6, 7 and 9 by around half; the remaining questions were less popular, and some received almost no "hits". Overall, the highest scoring questions (averaging over half-marks) were 1, 2 and 9, along with 13 (very few attempts, but those who braved it scored very well). This at least is indicative that candidates are being careful in exercising some degree of thought when choosing (at least the first four) 'good' questions for themselves, although finding six successful questions then turned out to be a key discriminating factor of candidates' abilities from the examining team's perspective. Each of questions 4-8, 11 & 12 were rather poorly scored on, with average scores of only 5.5 to 6.6.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.7
Banger Comparisons: 1
The end $A$ of an inextensible string $AB$ of length $\pi$
is attached to a point on the circumference
of a fixed circle of unit radius and
centre $O$. Initially the string is straight
and tangent to the circle. The string is then wrapped round the circle
until the end $B$ comes into
contact with the circle.
The string remains taut during the motion,
so that a section of the string is in contact with the circumference
and the remaining section is straight.
Taking $O$ to be the origin of cartesian coordinates with $A$ at $(-1,0)$
and $B$ initially at $(-1, \pi)$, show that the
curve described by $B$ is given parametrically by
\[
x= \cos t + t\sin t\,, \ \ \ \ \ \
y= \sin t - t\cos t\,,
\]
where $t$ is the angle shown in the diagram.
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\begin{pspicture*}(-5.4,-1)(7,7)
\pspolygon(-1.22,3.03)(-0.87,3.17)(-1.01,3.52)(-1.36,3.38)
\parametricplot{-0.17}{3.3}{1*3.64*cos(t)+0*3.64*sin(t)+0|0*3.64*cos(t)+1*3.64*sin(t)+0}
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\rput[tl](-0.45,-0.1){$O$}
\rput[tl](-4.12,0.46){$A$}
\rput[tl](6.11,6.8){$B$}
\rput[tl](0.25,0.6){$t$}
\psline{->}(-7.22,0)(5.78,0)
\psline{->}(0,-1.53)(0,6)
\rput[tl](-0.08,6.45){$y$}
\rput[tl](5.85,0.1){$x$}
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Find the value, $t_0$, of $t$ for which $x$
takes its maximum value on the curve,
and sketch the curve.
Use the area integral $\displaystyle \int y \frac{\d x}{\d t} \,
\d t\,$
to find the area between the curve and the $x$ axis
for~\hbox{$\pi \ge t \ge t_0$}.
Find the area swept out by the string (that is, the area between the
curve described by
$B$ and the semicircle shown in the diagram).
This was the least popular of the pure maths questions, probably with good reason, as it included a lengthy introduction and a diagram. In the first part, despite showing candidates that the point where the string leaves the circle is in the second quadrant, the necessary coordinate geometry work provided a considerable challenge. The second part, finding the maximum of x by standard differentiation techniques, proved to be relatively straightforward and a lot of candidates managed to get full marks for this work. The third part presented the core challenge of this question, in the sense that not many candidates seemed to have understood how to set the limits of the parametric integral, and 'benefit of the doubt' had to be fairly generously applied to those who switched signs when it suited them. The next part of the question involved applying integration by parts in order to evaluate the integrals but surprisingly few candidates managed to do so entirely successfully. Some of the common issues were the signs, that now needed to be fully consistent, and the application of parts twice after using double-angle formulae. The notion of the "total area swept out by the string" was also not so well understood, with only a very few realising that they needed to integrate from t = 0 to t = ½π as well. Most remembered to subtract the area of the semi-circle though.