Year: 2016
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Work, energy and Power 2
No solution available for this problem.
A substantially larger number of candidates took the paper this year: 14% more than in 2015. However, the mean score was virtually identical to that in 2015. Five questions were very popular, with two being attempted by in excess of 90% of the candidates, but once again, all questions were attempted by significant numbers, with only one dipping under 10% attempting it, and every question was answered perfectly by at least one candidate. Most candidates kept to six sensible attempts, although some did several more scoring weakly overall, except in six outstanding cases that earned very high marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1475.6
Banger Comparisons: 2
Three pegs $P$, $Q$ and $R$
are fixed on a smooth horizontal table in such a way that they
form the vertices of an equilateral triangle of side $2a$. A particle $X$
of mass $m$ lies on the table.
It is attached to the pegs by three springs, $PX$, $QX$ and $RX$, each
of modulus of elasticity
$\lambda$ and
natural length $l$,
where $l < \frac{ \ 2 }{\sqrt3}\, a$.
Initially the particle is in equilibrium.
Show that the extension in each spring is
$\frac{\ 2}{\sqrt3}\,a -l\,$.
The particle is then pulled a small distance directly towards
$P$ and released. Show that the tension $T$ in the spring
$RX$
is given by
\[
T= \frac {\lambda} l
\left( \sqrt{\frac {4a^2}3 + \frac{2ax}{\sqrt3} +x^2\; }\; -l\right)
,
\]
where $x$ is the displacement of $X$ from its equilibrium position.
Show further that the particle performs approximate
simple harmonic motion with period
\[
2\pi \sqrt{ \frac{4mla}{3 (4a-\sqrt3 \, l)\lambda } \; }\,.
\]
Although not overly popular, being attempted by less than a fifth of candidates, this question was moderately successful, a little better than question 3. Finding the extensions and tension in the first two parts of the question was completed by most candidates, but having generally written down the equation of motion, very few thought of applying the binomial expansion and so could not proceed to the final result.