OCR M2 (Mechanics 2) 2012 June

1 A particle, of mass 0.8 kg , moves along a smooth horizontal surface. It hits a vertical wall, which is at right angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of restitution between the particle and the wall is 0.3 . Find

1. the impulse that the wall exerts on the particle,
2. the kinetic energy lost in the impact.

2 A car of mass 1600 kg moves along a straight horizontal road. The resistance to the motion of the car has constant magnitude 800 N and the car's engine is working at a constant rate of 20 kW .

1. Find the acceleration of the car at an instant when the car's speed is \(20 \mathrm {~ms} ^ { - 1 }\). The car now moves up a hill inclined at \(4 ^ { \circ }\) to the horizontal. The car's engine continues to work at 20 kW and the magnitude of the resistance to motion remains at 800 N .
2. Find the greatest steady speed at which the car can move up the hill.

3
\includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-2_599_677_1151_696} A uniform beam \(A B\) of mass 15 kg and length 4 m is freely hinged to a vertical wall at \(A\). The beam is held in equilibrium in a horizontal position by a light rod \(P Q\) of length \(1.5 \mathrm {~m} . P\) is fixed to the wall vertically below \(A\) and \(P Q\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram). The force exerted on the beam at \(Q\) by the rod is in the direction \(P Q\). Find

1. the magnitude of the force exerted on the beam at \(Q\),
2. the magnitude and direction of the force exerted on the beam at \(A\).

4 A boy throws a small ball at a vertical wall. The ball is thrown horizontally, from a point \(O\), at a speed of \(14.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it hits the wall at a point which is 0.2 m below the level of \(O\).

1. Find the horizontal distance from \(O\) to the wall. The boy now moves so that he is 6 m from the wall. He throws the ball at an angle of \(15 ^ { \circ }\) above the horizontal. The ball again hits the wall at a point which is 0.2 m below the level from which it was thrown.
2. Find the speed at which the ball was thrown.

5 A particle \(P\), of mass 2 kg , is attached to fixed points \(A\) and \(B\) by light inextensible strings, each of length 2 m . \(A\) and \(B\) are 3.2 m apart with \(A\) vertically above \(B\). The particle \(P\) moves in a horizontal circle with centre at the mid-point of \(A B\).

1. Find the tension in each string when the angular speed of \(P\) is \(4 \mathrm { rads } ^ { - 1 }\).
2. Find the least possible speed of \(P\).

6 Three particles \(A , B\) and \(C\) are in a straight line on a smooth horizontal surface. The particles have masses \(0.2 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and 0.6 kg respectively. \(B\) is at rest. \(A\) is projected towards \(B\) with a speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) and collides with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after the collision is \(0.8 \mathrm {~ms} ^ { - 1 }\) and find the speed of \(A\) after the collision.
   \(C\) is moving with speed \(0.2 \mathrm {~ms} ^ { - 1 }\) in the same direction as \(B\). Particle \(B\) subsequently collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\).
2. Find the set of values for \(e\) such that \(B\) does not collide again with \(A\).

7
\includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-4_353_579_248_744} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium \(A B C D\) with \(A B\) and \(C D\) perpendicular to \(A D\). The lengths of \(A B\) and \(A D\) are each 5 cm and the length of \(C D\) is \(( a + 5 ) \mathrm { cm }\).

1. Show the distance of the centre of mass of the prism from \(A D\) is $$\frac { a ^ { 2 } + 15 a + 75 } { 3 ( a + 10 ) } \mathrm { cm } .$$ The prism is placed with the face containing \(A B\) in contact with a horizontal surface.
2. Find the greatest value of \(a\) for which the prism does not topple. The prism is now placed on an inclined plane which makes an angle \(\theta ^ { \circ }\) with the horizontal. \(A B\) lies along a line of greatest slope with \(B\) higher than \(A\).
3. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the greatest value of \(\theta\) for which the prism does not topple.