Year: 2012
Paper: 3
Question Number: 9
Course: zNo longer examinable
Section: Moments of inertia
No solution available for this problem.
The number of candidates attempting more than six questions was, as last year, about 25%, though most of these extra attempts achieved little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A pulley consists of a disc of radius $r$
with centre $O$
and a light thin axle through $O$ perpendicular
to the plane of the disc. The disc is non-uniform,
its mass is $M$ and its
centre of mass is at $O$. The
axle is fixed and horizontal.
Two particles, of masses $m_1$ and $m_2$ where $m_1>m_2$,
are connected by a light inextensible string which passes over
the pulley.
The contact between the string
and the pulley is rough enough to prevent the string sliding.
The pulley turns and the vertical force on the axle is
found, by measurement, to be~$P+Mg$.
\begin{questionparts}
\item
The moment of inertia of the pulley about its axle is calculated
assuming that the pulley rotates without friction about its axle.
Show that the calculated value is
\[
\frac{((m_1 + m_2)P - 4m_1m_2g)r^2}
{(m_1 + m_2)g - P}\,.
\tag{$*$}\]
\item
Instead, the moment of inertia of the pulley about its axle
is calculated
assuming that a couple of magnitude $C$ due to
friction acts on the axle of the pulley.
Determine whether
this calculated value is greater or
smaller than $(*)$.
Show that $C<(m_1-m_2)rg$.
\end{questionparts}
The second least popular question attempted by only a couple of dozen candidates with very little success, less than any other question. The problem was none of the candidates appreciated how to handle the algebra to obtain the first result, even if they had obtained the equations of motion. Unfortunately, they rarely had the full set of equations of motion. As a consequence, they made no progress on the second part.