Year: 2014
Paper: 3
Question Number: 10
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
A 10% increase in the number of candidates and the popularity of all questions ensured that all questions had a good number of attempts, though the first two questions were very much the most popular. Every question received at least one absolutely correct solution. In most cases when candidates submitted more than six solutions, the extra ones were rarely substantial attempts. Five sixths gave in at least six attempts.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1473.3
Banger Comparisons: 2
Two particles $X$ and $Y$, of equal mass $m$, lie on a
smooth horizontal table and are connected by a
light elastic spring of natural
length $a$ and modulus of elasticity $\lambda$. Two more springs,
identical to the first,
connect
$X$ to a point $P$ on the table and
$Y$
to a point $Q$ on the table. The distance between $P$ and $Q$ is $3a$.
Initially, the particles are held so that $XP=a$,
$YQ= \frac12 a\,$, and $PXYQ$ is a straight line.
The particles are then released.
At time $t$, the particle $X$ is a distance $a+x$ from $P$ and the
particle $Y$ is a distance $a+y$ from $Q$. Show that
\[
m \frac{\.d ^2 x}{\.d t^2} = -\frac\lambda a (2x+y)
\]
and find a similar expression involving $\dfrac{\.d^2 y}{\.d t^2}$.
Deduce that
\[
x-y = A\cos \omega t +B \sin\omega t
\]
where $A$ and $B$ are constants to be determined and
$ma\omega^2=\lambda$. Find a similar expression for $x+y$.
Show that $Y$ will never return to its initial position.
This was attempted by a quarter of the candidates with scores just below those achieved in question 5. Good candidates could obtain full marks in less than a page of working whilst weak ones spent a lot of effort trying to solve differential equations for x and y because they hadn't spotted the change of variables to u = x+y and v = x−y. The vast majority could obtain the first equation, often using the given result as a guide. However, there was frequent confusion between extension and total length of the springs. In addition, sign errors in v prevented the next part from working out. Lots did not realise to work with u and v, but those that saw SHM in one of these, saw it in the other. Likewise, with initial conditions, quite a few overlooked x(0) = 0, y(0) = 0, which prevented them solving for the constants, and also the sign was often overlooked in the condition ẋ = −ẏ. In attempting the last result, some used the factor formula, which worked but was unnecessary. Quite often, they stumbled over the final step of logic ending up with apparent contradictions such as √3 ≤ 1, which is of course false, but did not demonstrate full understanding.