OCR MEI M2 (Mechanics 2) 2015 June

1 A thin uniform rigid rod JK of length 1.2 m and weight 30 N is resting on a rough circular cylinder which is fixed to a floor. The axis of symmetry of the cylinder is horizontal and at all times the rod is perpendicular to this axis. Initially, the rod is horizontal and its point of contact with the cylinder is 0.4 m from K . It is held in equilibrium by resting on a small peg at J . This situation is shown in Fig. 1.1. \begin{figure}[h] \end{figure}

1. Calculate the force exerted by the peg on the rod and also the force exerted by the cylinder on the rod. A small object of weight \(W \mathrm {~N}\) is attached to the rod at K .
2. Find the greatest value of \(W\) for which the rod maintains its contact at J . The object at K is removed. Fig. 1.2 shows the rod resting on the cylinder with its end J on the floor, which is smooth and horizontal. The point of contact of the rod with the cylinder is 0.3 m from K. Fig. 1.2 also shows the normal reaction, \(S \mathrm {~N}\), of the floor on the rod, the normal reaction, \(R \mathrm {~N}\), of the cylinder on the rod and the frictional force \(F \mathrm {~N}\) between the cylinder and the rod. Suppose the rod is in equilibrium at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\theta < 90\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{71d839d8-12ca-4806-8f74-c572e7e21891-2_392_945_1578_561} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Find \(S\). Find also expressions in terms of \(\theta\) for \(R\) and \(F\). The coefficient of friction between the cylinder and the rod is \(\mu\).
4. Determine a relationship between \(\mu\) and \(\theta\).

2 Fig. 2 shows a wedge of angle \(30 ^ { \circ }\) fixed to a horizontal floor. Small objects P , of mass 8 kg , and Q , of mass 10 kg , are connected by a light inextensible string that passes over a smooth pulley at the top of the wedge. The part of the string between P and the pulley is parallel to a line of greatest slope of the wedge. Q hangs freely and at no time does either P or Q reach the pulley or P reach the floor. \begin{figure}[h] \end{figure}

1. Assuming the string remains taut, find the change in the gravitational potential energy of the system when Q descends \(h \mathrm {~m}\), stating whether it is a loss or a gain. Object P makes smooth contact with the wedge. The system is set in motion with the string taut.
2. Find the speed at which Q hits the floor if
   (A) the system is released from rest with Q a distance of 1.2 m above the floor,
   (B) instead, the system is set in motion with Q a distance of 0.3 m above the floor and P travelling down the slope at \(1.05 \mathrm {~ms} ^ { - 1 }\). The sloping face is roughened so that the coefficient of friction between object P and the wedge is 0.9 . The system is set in motion with the string taut and P travelling down the slope at \(2 \mathrm {~ms} ^ { - 1 }\).
3. How far does P move before it reaches its lowest point?
4. Determine what happens to the system after P reaches its lowest point.
5. Calculate the power of the frictional force acting on P in part (iii) at the moment the system is set in motion. \section*{Question 3 begins on page 4.}

3 A uniform heavy lamina occupies the region shaded in Fig. 3. This region is formed by removing a square of side 1 unit from a square of side \(a\) units (where \(a > 1\) ). \begin{figure}[h] \end{figure} Relative to the axes shown in Fig. 3, the centre of mass of the lamina is at \(( \bar { x } , \bar { y } )\).

1. Show that \(\bar { x } = \bar { y } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).
   [0pt] [You may need to use the result \(\frac { a ^ { 3 } - 1 } { 2 \left( a ^ { 2 } - 1 \right) } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).]
2. Show that the centre of mass of the lamina lies on its perimeter if \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\). In another situation, \(a = 4\).
   A particle of mass one third that of the lamina is attached to the lamina at vertex B ; the lamina with the particle is freely suspended from vertex A and hangs in equilibrium. The positions of A and B are shown in Fig. 3.
3. Calculate the angle that AB makes with the vertical.

4

1. Two discs, P of mass 4 kg and Q of mass 5 kg , are sliding along the same line on a smooth horizontal plane when they collide. The velocity of P before the collision and the velocity of Q after the collision are shown in Fig. 4. P loses \(\frac { 5 } { 9 }\) of its kinetic energy in the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{71d839d8-12ca-4806-8f74-c572e7e21891-5_294_899_390_584} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Show that after the collision P has a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to its original motion. While colliding, the discs are in contact for \(\frac { 1 } { 5 } \mathrm {~s}\).
   2. Find the impulse on P in the collision and the average force acting on the discs.
   3. Find the velocity of Q before the collision and the coefficient of restitution between the two discs.
2. A particle is projected from a point 2.5 m above a smooth horizontal plane. Its initial velocity is \(5.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) below the horizontal, where \(\sin \theta = \frac { 15 } { 17 }\). The coefficient of restitution between the particle and the plane is \(\frac { 4 } { 5 }\).
   1. Show that, after bouncing off the plane, the greatest height reached by the particle is 2.5 m .
   2. Calculate the horizontal distance between the two points at which the particle is 2.5 m above the plane.