Standard +0.8 This is a multi-step statics problem requiring resolution of forces in two directions, taking moments about a strategic point, and finding the limiting friction condition. While it involves several standard techniques (resolving forces, moments, friction), the geometry with the perpendicular string and the need to coordinate multiple equations to find μ makes it moderately challenging but still within typical Further Maths scope.
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\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-2_684_714_246_717}
A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(4 a\), rests with the end \(A\) on rough horizontal ground. The point \(C\) on \(A B\) is such that \(A C = 3 a\). A light inextensible string has one end attached to the point \(P\) which is at a distance \(5 a\) vertically above \(A\), and the other end attached to \(C\). The rod and the string are in the same vertical plane and the system is in equilibrium with angle \(A C P\) equal to \(90 ^ { \circ }\) (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). Show that the least possible value of \(\mu\) is \(\frac { 24 } { 43 }\).
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\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-2_684_714_246_717}
A uniform $\operatorname { rod } A B$, of mass $m$ and length $4 a$, rests with the end $A$ on rough horizontal ground. The point $C$ on $A B$ is such that $A C = 3 a$. A light inextensible string has one end attached to the point $P$ which is at a distance $5 a$ vertically above $A$, and the other end attached to $C$. The rod and the string are in the same vertical plane and the system is in equilibrium with angle $A C P$ equal to $90 ^ { \circ }$ (see diagram). The coefficient of friction between the rod and the ground is $\mu$. Show that the least possible value of $\mu$ is $\frac { 24 } { 43 }$.
\hfill \mbox{\textit{CAIE FP2 2013 Q1 [8]}}