Year: 2013
Paper: 3
Question Number: 9
Course: UFM Pure
Section: Second order differential equations
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: 1484.8
Banger Comparisons: 1
A sphere of radius $R$ and uniform density $\rho_{\text{s}}$ is floating in
a large tank of liquid of uniform density $\rho$.
Given that the centre of the sphere is a distance
$x$ above the level of the liquid, where $x < R$, show that the volume
of liquid displaced is
\[
\frac \pi 3 (2R^3-3R^2x +x^3)\,.
\]
The sphere is acted upon by two forces only: its weight and
an upward force equal in magnitude to the weight of the liquid it has
displaced. Show that
\[
4 R^3\rho_{\text{s}} (g+\ddot x) = (2R^3 -3R^2x +x^3)\rho g\,.
\]
Given that the sphere is in equilibrium when $x=\frac12 R$, find
$\rho_{\text{s}}$ in terms of $\rho$. Find, in terms of $R$ and $g$,
the period of small oscillations
about this equilibrium position.
About a fifth attempted this, with the same success as question 7. Common errors were false attempts for the volume at the beginning using hemisphere and cones, and in the last part approximating θ small rather than the correct small angle. Many candidates were successful as far as the equilibrium but couldn't deal with the small oscillations successfully.