Year: 2014
Paper: 2
Question Number: 11
Course: LFM Pure and Mechanics
Section: Motion on a slope
No solution available for this problem.
There were good solutions presented to all of the questions, although there was generally less success in those questions that required explanations of results or the use of diagrams and graphs to reach the solution. Algebraic manipulation was generally well done by many of the candidates although a range of common errors such as confusing differentiation and integration and simple arithmetic slips were evident. Candidates should also be advised to use the methods that are asked for in questions unless it is clear that other methods will be accepted (such as by the use of the phrase "or otherwise").
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1504.7
Banger Comparisons: 2
A small smooth ring $R$ of mass $m$ is free to slide on a fixed smooth
horizontal rail. A light inextensible string of length~$L$ is
attached to one end,~$O$, of the rail. The string passes through
the ring, and a particle~$P$ of mass~$km$ (where $k>0$)
is attached to its other
end; this part of the string hangs at an acute
angle $\alpha$ to the vertical and
it is given that $\alpha$ is constant in the motion.
Let $x$ be the distance between $O$ and the ring. Taking the
$y$-axis to be vertically upwards, write down the Cartesian
coordinates of~$P$ relative to~$O$ in terms of $x$, $L$
and~$\alpha$.
\begin{questionparts}
\item
By considering the vertical component of the equation of motion of $P$,
show that
\[
km\ddot x \cos\alpha = T \cos\alpha - kmg\,,
\]
where $T$ is the tension in the string. Obtain two similar equations
relating to the horizontal components of the equations of motion of
$P$ and $R$.
\item Show that
$\dfrac {\sin\alpha}{(1-\sin\alpha)^2_{\vphantom|}} = k$, and
deduce, by means of a sketch or otherwise, that motion with $\alpha$
constant
is possible for all values of~$k$.
\item Show that $\ddot x = -g\tan\alpha\,$.
\end{questionparts}
Many candidates who attempted this question struggled, particularly due to a difficulty in drawing a diagram to represent the situation. From these incorrect diagrams candidates often reached results where one of the signs did not match that given in the question. The calculation of the acceleration was found to be difficult by many of the candidates, although those who understood that differentiation of the coordinates of P would give the acceleration were then able to complete the rest of the question correctly. Those candidates that attempted the final part of the question were able to solve it correctly.