Challenging +1.8 This is a challenging M2 statics problem requiring careful geometric analysis to find the position of contact point T, resolution of forces in two directions, friction conditions at two surfaces, and taking moments. The semicircular prism geometry adds complexity beyond standard rod-on-peg questions, requiring students to determine the normal direction at T and apply limiting friction μR at both contacts simultaneously. The 8-mark allocation reflects substantial multi-step reasoning, though the techniques themselves are standard M2 content.
9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\).
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475}
The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
Find the value of \(\mu\). [0pt]
[8 marks]
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}
9 A uniform rod, $P Q$, of length $2 a$, rests with one end, $P$, on rough horizontal ground and a point $T$ resting on a rough fixed prism of semicircular cross-section of radius $a$, as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both $P$ and $T$ is $\mu$.\\
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475}
The rod is on the point of slipping when it is inclined at an angle of $30 ^ { \circ }$ to the horizontal.
Find the value of $\mu$.\\[0pt]
[8 marks]\\
\includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}
\hfill \mbox{\textit{AQA M2 2015 Q9 [8]}}