Year: 2016
Paper: 3
Question Number: 10
Course: UFM Mechanics
Section: Circular Motion 2
No solution available for this problem.
A substantially larger number of candidates took the paper this year: 14% more than in 2015. However, the mean score was virtually identical to that in 2015. Five questions were very popular, with two being attempted by in excess of 90% of the candidates, but once again, all questions were attempted by significant numbers, with only one dipping under 10% attempting it, and every question was answered perfectly by at least one candidate. Most candidates kept to six sensible attempts, although some did several more scoring weakly overall, except in six outstanding cases that earned very high marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A smooth plane is inclined
at an angle $\alpha$ to the horizontal. A particle $P$ of mass $m$
is attached to a fixed point $A$
above the plane by a light inextensible string of length $a$.
The particle rests in equilibrium on the plane,
and the string makes an angle $\beta$ with the plane.
The particle is given a horizontal impulse parallel
to the plane so that it has an initial speed
of $u$.
Show that the particle will not immediately leave the plane
if $ag\cos(\alpha + \beta)> u^2 \tan\beta$.
Show further that a necessary condition for the
particle to perform a complete circle whilst in
contact with the plane is
$6\tan\alpha \tan \beta < 1$.
One of the least popular questions, being attempted by about a seventh of the candidates, it was the least successfully answered with just under quarter marks scored. As ever, there were some very strong solutions. Generally, the first result of the question was done well as they could resolve accurately and identify the condition, v > 0. For the final result, not many identified the condition v > 0, and so could not proceed. Generally, energy was conserved well, though some omitted one or other of the kinetic energy terms. As far as circular motion was concerned, some treated the radius as r rather than r sin α.