Rod on smooth peg or cylinder

\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find

1. the force acting on the rod at \(P\), [3]
2. the coefficient of friction between the ground and the rod. [5]

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]

\includegraphics{figure_1} A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5a\). A uniform rod \(AB\), of length \(16a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(AC = 12a\) and angle \(CAO = \alpha\), as shown in Figure 1. Points \(A\), \(C\), \(B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(AO = 13a\) [1]

The normal reaction on the rod at \(C\) has magnitude \(kW\)

1. Show that \(k = \frac{8}{13}\) [3]

The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta°\) to the horizontal.

1. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\)
   [8]