Year: 2021
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Moments
No solution available for this problem.
Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Two particles, of masses $m_1$ and $m_2$ where $m_1 > m_2$, are attached to the ends of a light, inextensible string. A particle of mass $M$ is fixed to a point $P$ on the string. The string passes over two small, smooth pulleys at $Q$ and $R$, where $QR$ is horizontal, so that the particle of mass $m_1$ hangs vertically below $Q$ and the particle of mass $m_2$ hangs vertically below~$R$. The particle of mass $M$ hangs between the two pulleys with the section of the string $PQ$ making an acute angle of $\theta_1$ with the upward vertical and the section of the string $PR$ making an acute angle of $\theta_2$ with the upward vertical. $S$ is the point on $QR$ vertically above~$P$. The system is in equilibrium.
\begin{questionparts}
\item Using a triangle of forces, or otherwise, show that:
\begin{enumerate}[label=(\alph*)]
\item $\sqrt{m_1^2 - m_2^2} < M < m_1 + m_2$\,;
\item $S$ divides $QR$ in the ratio $r : 1$, where
\[
r = \frac{M^2 - m_1^2 + m_2^2}{M^2 - m_2^2 + m_1^2}.
\]
\end{enumerate}
\item You are now given that $M^2 = m_1^2 + m_2^2$.
Show that $\theta_1 + \theta_2 = 90^\circ$ and determine the ratio of $QR$ to $SP$ in terms of the masses only.
\end{questionparts}
It was pleasing to see that many candidates chose to draw a diagram to represent the setup of the problem. Solutions to part (i) were often good, but marks were often lost due to lack of justification, both for the triangle inequality and for reasoning involving acute angles. Candidates often also failed to equate the tension on either side of the pulleys. Many candidates attempted part (ii) having failed to complete the previous and most were able to obtain a mark here by using the results that had been given in the previous part.