Year: 2013
Paper: 3
Question Number: 11
Course: UFM Mechanics
Section: Work, energy and Power 2
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: 1484.0
Banger Comparisons: 1
An equilateral triangle, comprising three light rods each of
length $\sqrt3a$, has a particle of mass $m$ attached to each of
its vertices. The triangle is suspended horizontally from a point vertically
above its centre by three identical springs, so that the springs and
rods form a tetrahedron.
Each spring has natural length $a$ and modulus
of elasticity $kmg$, and is light.
Show that when the springs make an angle $\theta$
with the horizontal the tension in each spring is
\[
\frac{ kmg(1-\cos\theta)}{\cos\theta}\,.
\]
Given that the triangle is in equilibrium when $\theta = \frac16 \pi$, show
that $k=4\sqrt3 +6$.
The triangle is released from rest from the position
at which $\theta=\frac13\pi$.
Show that when it passes through the equilibrium position its speed $V$
satisfies
\[
V^2 = \frac{4ag}3(6+\sqrt3)\,.
\]
A fifth of the candidates attempted this question, with marginally less success than question 3. Most that attempted this question managed to achieve the first two results successfully, unless they got the diagram wrong. However, the final result was found trickier as some forgot to include the gravitational potential energy, some failed to evaluate the correct elastic potential energy and there were many mistakes made handling the surds.