Year: 2023
Paper: 3
Question Number: 10
Course: UFM Mechanics
Section: Moments
No solution available for this problem.
The total entry was a marginal increase on that of 2022 (by just over 1%). Two questions were attempted by more than 90% of candidates, another two by 80%, and another two by about two thirds. The least popular questions were attempted by more than a sixth of candidates. All the questions were perfectly answered by at least three candidates (but mostly more than this), with one being perfectly answered by eighty candidates. Very nearly 90% of candidates attempted no more than 7 questions. One general comment regarding all the questions is that candidates need to make sure that they read the question carefully, paying particular attention to command words such as "hence" and "show that".
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A thin uniform beam $AB$ has mass $3m$ and length $2h$. End $A$ rests on rough horizontal ground and the beam makes an angle of $2\beta$ to the vertical, supported by a light inextensible string attached to end $B$. The coefficient of friction between the beam and the ground at $A$ is $\mu$.
The string passes over a small frictionless pulley fixed to a point $C$ which is a distance $2h$ vertically above $A$. A particle of mass $km$, where $k < 3$, is attached to the other end of the string and hangs freely.
\begin{questionparts}
\item Given that the system is in equilibrium, find an expression for $k$ in terms of $\beta$ and show that $k^2 \leqslant \dfrac{9\mu^2}{\mu^2 + 1}$.
\item A particle of mass $m$ is now fixed to the beam at a distance $xh$ from $A$, where $0 \leqslant x \leqslant 2$. Given that $k = 2$, and that the system is in equilibrium, show that
\[\frac{F^2}{N^2} = \frac{x^2 + 6x + 5}{4(x+2)^2}\,,\]
where $F$ is the frictional force and $N$ is the normal reaction on the beam at $A$.
By considering $\dfrac{1}{3} - \dfrac{F^2}{N^2}$, or otherwise, find the minimum value of $\mu$ for which the beam can be in equilibrium whatever the value of $x$.
\end{questionparts}
The most popular of the Applied questions, it was also the least successfully attempted, and was only slightly better attempted than question 4. If a candidate found this question difficult, it tended to be from the start, failing to draw a correct diagram of what was going on. If they did set up the problem correctly then finding different angles in terms of the given angle β proved problematic. This meant that often sin was found instead of cos and vice versa, within attempted solutions. Many successfully took moments about A and resolved forces vertically and horizontally, but most were unable to use these to produce the inequality required in part (i). Those few that did well in part (i) generally also did well in part (ii), with only one adjustment to the diagram needed, and the resulting algebra was worked through with little issue. The hint for the final part was used well by candidates, but only a few managed to turn the resulting square into a minimum value for μ.