Challenging +1.8 This is a challenging statics problem requiring careful geometric analysis to find the contact point on a circular prism, then applying equilibrium conditions (forces and moments) with friction at two surfaces simultaneously. The 'point of slipping' condition means friction is limiting at both contacts, adding complexity. While the geometry is tractable with the 30° angle, coordinating all constraints requires systematic problem-solving beyond routine textbook exercises.
A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular 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{figure_6}
The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal.
Find the value of \(\mu\).
[8]
A uniform rod, PQ, of length $2a$, rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular 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{figure_6}
The rod is on the point of slipping when it is inclined at an angle of 30$^0$ to the horizontal.
Find the value of $\mu$.
[8]
\hfill \mbox{\textit{SPS SPS FM Mechanics 2022 Q6 [8]}}