Year: 2013
Paper: 3
Question Number: 10
Course: zNo longer examinable
Section: Moments of inertia
No solution available for this problem.
With the number of candidates submitting scripts up by some 8% from last year, and whilst inevitably some questions were more popular than others, namely the first two, 7 then 4 and 5 to a lesser extent, all questions on the paper were attempted by a significant number of candidates. About a sixth of candidates gave in answers to more than six questions, but the extra questions were invariably scoring negligible marks. Two fifths of the candidates gave in answers to six questions.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A uniform rod $AB$ has mass $M$ and length $2a$. The point
$P$ lies on the rod a distance $a-x$ from~$A$. Show that the
moment of inertia of the rod about an axis through $P$ and
perpendicular to
the rod is
\[
\tfrac13 M(a^2 +3x^2)\,.
\]
The rod is free
to rotate, in a horizontal plane, about a fixed vertical axis through $P$.
Initially the rod is at rest. The end $B$ is struck by a particle of
mass $m$ moving horizontally with speed $u$ in a direction
perpendicular to the rod.
The coefficient of restitution between the rod and the particle is $e$.
Show that the angular velocity of the rod immediately after impact
is
\[
\frac{3mu(1+e)(a+x)}{M(a^2+3x^2) +3m(a+x)^2}\,.
\]
In the case $m=2M$, find the value of $x$ for which the angular velocity
is greatest and show that this angular velocity is $u(1+e)/a\,$.
The number of candidates attempting this was almost identical to that attempting question 3 with marginally more success making it the third best attempted question. Most obtained the moment of inertia correctly, and many found the angular velocity correctly. Provided that they had correctly applied conservation of angular momentum, and Newton's law of elasticity, they almost all worked out the required result. Some attempted to use conservation of linear momentum whilst others did not use conservation of angular momentum correctly. Most then knew how to differentiate, but many made computation errors. Even if they got the correct quadratic equation at the end, many solved it wrongly. Very few showed that the feasible solution did indeed generate a maximum.