Year: 2008
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Variable Force
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A particle of mass $m$ is initially at rest
on a rough horizontal surface. The particle experiences a force
$mg\sin \pi t$, where $t$ is time, acting in
a fixed horizontal direction.
The coefficient of friction between the particle and the surface is $\mu$.
Given that the particle starts to
move first at $t=T_0$, state the relation between
$T_0$ and $\mu$.
\begin{questionparts}
\item For $\mu = \mu_0$,
the particle comes to rest for the first time at $t=1$.
Sketch the acceleration-time graph for $0\le t \le 1$. Show that
\[
1+\left(1-\mu_0^2\right)^{\frac12}
-\mu_0\pi +\mu_0 \arcsin \mu_0 =0\,.
\]
\item For $\mu=\mu_0$
sketch the acceleration-time graph for $0\le t\le 3$.
Describe the motion of the particle in this case and in the case $\mu=0$.
\end{questionparts}
\noindent[{\bf Note:} $\arcsin x$ is another notation for
$\sin^{-1}x$.\ ]
Of the three Mechanics questions, this was the most popular with just under a quarter of the candidates attempting it, but with least success. In spite of obtaining the relation in the stem of the question, many failed to appreciate its consequence for the acceleration-time graph in part (i) and as a consequence made little further progress. If candidates managed part (i), then they tended to complete the question barring minor errors, and the occasional assumption that the final simple case was simple harmonic motion.