Year: 2009
Paper: 3
Question Number: 10
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A light spring is fixed at its lower end and its axis is
vertical. When a certain particle $P$ rests on the top of the
spring, the compression is $d$. When, instead, $P$ is dropped onto the
top of the spring from a
height $h$ above it, the compression at time $t$ after $P$ hits the
top of the spring is $x$. Obtain a second-order differential
equation relating $x$ and $t$ for $0\le t \le T$, where $T$ is the time
at which $P$ first loses contact with the spring.
Find the solution of this equation in the form
\[
x= A + B\cos (\omega t) + C\sin(\omega t)\,,
\]
where the constants $A$, $B$, $C$ and $\omega$ are to be given in terms
of $d$, $g$ and $h$ as appropriate.
Show that
\[
T
= \sqrt{d/g\;} \left (2 \pi - 2 \arctan
\sqrt{2h/d\;}\;\right)\,.
\]
The second least popular question on the paper attempted by a twelfth of the entry, the success rate on it was comparable to questions 7 and 8. Mostly, candidates successfully found and solved the differential equation, but finding T stumped them, with alternative approaches successfully used in a very small number of cases.