CAIE M2 (Mechanics 2) 2011 November

1
\includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Calculate \(T\).
2. Find the least possible value of the coefficient of friction at \(A\).

2
\includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_525_913_1123_616} A particle \(P\) is projected from a point \(O\) at an angle of \(60 ^ { \circ }\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45 ^ { \circ }\) (see diagram).

1. Show that the speed of projection of \(P\) is \(8.20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. Calculate the time after projection when the direction of motion of \(P\) is \(45 ^ { \circ }\) above the horizontal.

4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.

1. Find the value of \(\theta\) at which the cone is about to topple.
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-3_476_1211_836_466}
3. Given that the solid immediately topples, find the least possible value of \(h\).