Year: 2022
Paper: 3
Question Number: 10
Course: UFM Mechanics
Section: Work, energy and Power 1
No solution available for this problem.
One question was attempted by well over 90% of the candidates two others by about 90%, and a fourth by over 80%. Two questions were attempted by about half the candidates and a further three questions by about a third of the candidates. Even the other three received attempts from a sixth of the candidates or more, meaning that even the least popular questions were markedly more popular than their counterparts in previous years. Nearly 90% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Two light elastic springs each have natural length $a$. One end of each spring is attached to a particle $P$ of weight $W$. The other ends of the springs are attached to the end-points, $B$ and $C$, of a fixed horizontal bar $BC$ of length $2a$. The moduli of elasticity of the springs $PB$ and $PC$ are $s_1 W$ and $s_2 W$ respectively; these values are such that the particle $P$ hangs in equilibrium with angle $BPC$ equal to $90^\circ$.
\begin{questionparts}
\item Let angle $PBC = \theta$. Show that $s_1 = \dfrac{\sin\theta}{2\cos\theta - 1}$ and find $s_2$ in terms of $\theta$.
\item Take the zero level of gravitational potential energy to be the horizontal bar $BC$ and let the total potential energy of the system be $-paW$. Show that $p$ satisfies
\[ \frac{1}{2}\sqrt{2} \geqslant p > \frac{1}{4}(1+\sqrt{3}) \]
and hence that $p = 0.7$, correct to one significant figure.
\end{questionparts}
Along with question 12, this was the least popular question on the paper with a sixth of candidates trying it, and scoring one third marks, on average. Part (i) was done well, demonstrating good use of Hooke's law, and resolving forces. It was failing to think about right angled triangle trigonometry that created most problems. In part (ii), many candidates got the signs of their potential energies wrong. Of those candidates who got to the correct expression for p most were able to find the maximum value correctly but very few were able to explain why the physical situation resulted in a restricted domain for the function. Showing the value of p must be 0.7 to one significant figure was rarely done well as many candidates used known approximations to the given surds without justifying the accuracy of these approximations.