Year: 2015
Paper: 3
Question Number: 9
Course: UFM Mechanics
Section: Variable Force
No solution available for this problem.
A very similar number of candidates to 2014 once again ensured that all questions received a decent number of attempts, with seven questions being very popular rather than five being so in 2014, but the most popular questions were attempted by percentages in the 80s rather than 90s. All but one question was answered perfectly at least once, the one exception receiving a number of very close to perfect solutions. About 70% attempted at least six questions, and in those cases where more than six were attempted, the extra attempts were usually fairly superficial.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1541.9
Banger Comparisons: 5
A particle $P$ of mass $m$ moves on a smooth fixed straight horizontal rail and is attached to a fixed peg $Q$ by a light elastic string of natural length $a$ and modulus $\lambda$. The peg $Q$ is a distance $a$ from the rail. Initially $P$ is at rest with $PQ=a$.
An impulse imparts to $P$ a speed $v$ along the rail. Let $x$ be the displacement at time $t$ of $P$ from its initial position. Obtain the equation
\[
\dot x^2 = v^2 - k^2 \left( \sqrt{x^2+a^2} -a\right)^{\!2}
\]
where $ k^2 = \lambda/(ma)$, $k>0$ and the dot denotes differentiation with respect to $t$.
Find, in terms of $k$, $a$ and $v$, the greatest value, $x_0$, attained by $x$. Find also the acceleration of $P$ at $x=x_0$.
Obtain, in the form of an integral, an expression for the period of the motion. Show that in the case $v\ll ka$ (that is, $v$ is much less than $ka$), this is approximately
\[
\sqrt {\frac {32a}{kv}}
\int_0^1 \frac 1 {\sqrt{1-u^4}} \, \d u \, .
\]
Just over 20% attempted this, making it the most popular non-Pure question, and attempts at it were slightly less successful than question 1. Quite a few found the first required equation from applying Newton's 2nd Law, when some made sign errors through not being careful with directions, and then integrating rather than from conserving energy. The expression for the velocity was found easily by the majority, and the expression for the acceleration was commonly still by Newton, though a lot of marks were lost by not substituting in. The last part was poorly done in general with few getting more than an opening line, and if they made a sensible substitution, very few expanded it correctly. Not many even attempted the last part of the question.