CAIE M2 (Mechanics 2) 2012 November

2
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\),
2. the least possible value of the coefficient of friction at \(B\).

4
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_538_885_1809_628} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45 ^ { \circ }\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius 0.67 m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\).

6
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-3_584_862_575_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).

1. Show that the centre of mass of the lamina lies on \(O B\).
2. Calculate the distance of the centre of mass of the lamina from \(O\).