OCR M2 (Mechanics 2) 2016 June

1 A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N . At a certain instant the car is accelerating at \(0.3 \mathrm {~ms} ^ { - 2 }\) and the engine of the car is working at a rate of 23 kW .

1. Find the speed of the car at this instant. Subsequently the car moves up a hill inclined at \(10 ^ { \circ }\) to the horizontal at a steady speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is still a constant 600 N .
2. Calculate the power of the car's engine as it moves up the hill.
   \(2 A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55 ^ { \circ }\) to the horizontal. \(A\) is below the level of \(B\) and \(A B = 4 \mathrm {~m}\). A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between the plane and \(P\) is 0.15 . Find
3. the work done against the frictional force as \(P\) moves from \(A\) to \(B\),
4. the initial speed of \(P\) at \(A\).

3 \begin{figure}[h] \end{figure} A uniform lamina \(A B D C\) is bounded by two semicircular arcs \(A B\) and \(C D\), each with centre \(O\) and of radii \(3 a\) and \(a\) respectively, and two straight edges, \(A C\) and \(D B\), which lie on the line \(A O B\) (see Fig. 1).

1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac { 13 a } { 3 \pi }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-3_1306_572_207_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(A B\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40 ^ { \circ }\) below the horizontal (see Fig. 2).
2. Calculate the tension in the string.
3. Find the direction of the force acting on the lamina at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-4_848_1491_251_287} A smooth solid cone of semi-vertical angle \(60 ^ { \circ }\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30 ^ { \circ }\) (see diagram).
4. Show that the magnitude of the contact force between the cone and the particle is \(\frac { 1 } { 6 } m \left( 2 \sqrt { 3 } g - 3 a \omega ^ { 2 } \right)\).
5. Given that \(a = 0.5 \mathrm {~m}\) and \(m = 3.5 \mathrm {~kg}\), find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string.

5 A uniform ladder \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 12 } { 13 }\). A man of weight 6 W is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 5 W } { 24 } \left( 1 + \frac { 6 x } { a } \right)\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\).
2. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight 6 W can now stand at the top of the ladder \(B\) without the ladder slipping.
3. Find the least possible value of \(\tan \alpha\).

6 The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(10 \mathrm {~ms} ^ { - 1 }\) before they collide. The kinetic energy lost due to the collision is 121.5 J .

1. Find the speed and direction of motion of each particle after the collision.
2. Find the coefficient of restitution between \(A\) and \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle \(P\) is projected with speed \(32 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac { 24 } { 25 }\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
3. Calculate the height of \(C\) above the ground and the distance \(A B\). Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
4. Given that the mass of \(P\) is 3 kg , find the magnitude and direction of the impulse exerted on \(P\) by the ground. The coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\).
5. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25 ^ { \circ }\) below the horizontal.