CAIE M2 (Mechanics 2) 2011 November

1
\includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_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]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_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.

3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.25 \mathrm {~kg} . P\) hangs in equilibrium below \(O\).

1. Calculate the distance \(O P\). The particle \(P\) is raised, and is released from rest at \(O\).
2. Calculate the speed of \(P\) when it passes through the equilibrium position.
3. Calculate the greatest value of the distance \(O P\) in the subsequent motion.

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]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
3. Given that the solid immediately topples, find the least possible value of \(h\).

5 A ball of mass 0.05 kg is released from rest at a height \(h \mathrm {~m}\) above the ground. At time \(t \mathrm {~s}\) after its release, the downward velocity of the ball is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance opposes the motion of the ball with a force of magnitude \(0.01 \nu \mathrm {~N}\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 0.2 v\). Hence find \(v\) in terms of \(t\).
2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\).

6 A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m . One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is \(T \mathrm {~N}\), and the bead rotates with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m , calculate \(\omega\), and hence find the kinetic energy of \(B\).
2. Given instead that angle \(A B C = 90 ^ { \circ }\), and that \(A B\) makes an angle \(\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\) with the vertical, calculate \(T\) and \(\omega\).