OCR M2 (Mechanics 2) 2013 June

2 The power developed by the engine of a car as it travels at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road is 20 kW .

1. Calculate the resistance to the motion of the car. The car, of mass 1500 kg , now travels down a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is unchanged.
2. Find the power produced by the engine of the car when the car has speed \(32 \mathrm {~ms} ^ { - 1 }\) and is accelerating at \(0.1 \mathrm {~ms} ^ { - 2 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-2_542_638_1208_717} A uniform semicircular arc \(A C B\) is freely pivoted at \(A\). The arc has mass 0.3 kg and is held in equilibrium by a force of magnitude \(P\) N applied at \(B\). The line of action of this force lies in the same plane as the arc, and is perpendicular to \(A B\). The diameter \(A B\) has length 4 cm and makes an angle of \(\theta ^ { \circ }\) with the downward vertical (see diagram).

1. Given that \(\theta = 0\), find the magnitude of the force acting on the arc at \(A\).
2. Given instead that \(\theta = 30\), find the value of \(P\).

4 A solid uniform cone has height 8 cm , base radius 5 cm and mass 4 kg . A uniform conical shell has height 10 cm , base radius 5 cm and mass 0.4 kg . The two shapes are joined together so that the circumferences of their circular bases coincide.

1. Find the distance of the centre of mass of the shape from the common circular base.
   \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-3_974_1141_484_463} The object is suspended with a string attached to the vertex of the cone and another string attached to the vertex of the conical shell. The object is in equilibrium with the strings vertical and the axis of symmetry of the object horizontal (see diagram).
2. Find the tension in each string.

5 A vertical hollow cylinder of radius 0.4 m is rotating about its axis. A particle \(P\) is in contact with the rough inner surface of the cylinder. The cylinder and \(P\) rotate with the same constant angular speed. The coefficient of friction between \(P\) and the cylinder is \(\mu\).

1. Given that the angular speed of the cylinder is \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) is on the point of moving downwards, find the value of \(\mu\). The particle is now attached to one end of a light inextensible string of length 0.5 m . The other end is fixed to a point \(A\) on the axis of the cylinder (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_681_970_660_536}
2. Find the angular speed for which the contact force between \(P\) and the cylinder becomes zero.

6
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_243_1179_1580_443} The masses of two particles \(A\) and \(B\) are 0.2 kg and \(m \mathrm {~kg}\) respectively. The particles are moving with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(u \mathrm {~ms} ^ { - 1 }\) in the same horizontal line and in the same direction (see diagram). The two particles collide and the coefficient of restitution between the particles is \(e\). After the collision, \(A\) and \(B\) continue in the same direction with speeds \(4 \left( 1 - e + e ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.

1. Find \(u\) and \(m\) in terms of \(e\).
2. Find the value of \(e\) for which the speed of \(A\) after the collision is least and find, in this case, the total loss in kinetic energy due to the collision.
3. Find the possible values of \(e\) for which the magnitude of the impulse that \(B\) exerts on \(A\) is 0.192 Ns .
   \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-5_744_887_264_589} The diagram shows a surface consisting of a horizontal part \(O A\) and a plane \(A B\) inclined at an angle of \(70 ^ { \circ }\) to the horizontal. A particle is projected from the point \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal \(O A\). The particle hits the plane \(A B\) at the point \(P\), with speed \(14 \mathrm {~ms} ^ { - 1 }\) and at right angles to the plane, 1.4 s after projection.
4. Show that the value of \(u\) is 15.9 , correct to 3 significant figures, and find the value of \(\theta\).
5. Find the height of \(P\) above the level of \(A\). The particle rebounds with speed \(v \mathrm {~ms} ^ { - 1 }\). The particle next lands at \(A\).
6. Find the value of \(v\).
7. Find the coefficient of restitution between the particle and the plane at \(P\).