8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder.
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\caption{Fig. 8}
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Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall.
The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.
- Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule.
The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
- By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
- By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).