OCR M1 (Mechanics 1) 2013 January

1 Three horizontal forces, acting at a single point, have magnitudes \(12 \mathrm {~N} , 14 \mathrm {~N}\) and 5 N and act along bearings \(000 ^ { \circ } , 090 ^ { \circ }\) and \(270 ^ { \circ }\) respectively. Find the magnitude and bearing of their resultant.

2 A particle \(P\) moves in a straight line. The displacement of \(P\) from a fixed point on the line is \(\left( t ^ { 4 } - 2 t ^ { 3 } + 5 \right) \mathrm { m }\), where \(t\) is the time in seconds. Show that, when \(t = 1.5\),

1. \(P\) is at instantaneous rest,
2. the acceleration of \(P\) is \(9 \mathrm {~ms} ^ { - 2 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-2_300_501_799_790} A particle \(P\) of mass 0.25 kg moves upwards with constant speed \(u \mathrm {~ms} ^ { - 1 }\) along a line of greatest slope on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The pulling force acting on \(P\) has magnitude \(T \mathrm {~N}\) and acts at an angle of \(20 ^ { \circ }\) to the line of greatest slope (see diagram). Calculate

1. the value of \(T\),
2. the magnitude of the contact force exerted on \(P\) by the plane. The pulling force \(T \mathrm {~N}\) acting on \(P\) is suddenly removed, and \(P\) comes to instantaneous rest 0.4 s later.
3. Calculate \(u\).

4 The acceleration of a particle \(P\) moving in a straight line is \(\left( t ^ { 2 } - 9 t + 18 \right) \mathrm { ms } ^ { - 2 }\), where \(t\) is the time in seconds.

1. Find the values of \(t\) for which the acceleration is zero.
2. It is given that when \(t = 3\) the velocity of \(P\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(t = 0\).
3. Show that the direction of motion of \(P\) changes before \(t = 1\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-3_462_405_258_845} A small smooth pulley is suspended from a fixed point by a light chain. A light inextensible string passes over the pulley. Particles \(P\) and \(Q\), of masses 0.3 kg and \(m \mathrm {~kg}\) respectively, are attached to the opposite ends of the string. The particles are released from rest at a height of 0.2 m above horizontal ground with the string taut; the portions of the string not in contact with the pulley are vertical (see diagram). \(P\) strikes the ground with speed \(1.4 \mathrm {~ms} ^ { - 1 }\). Subsequently \(P\) remains on the ground, and \(Q\) does not reach the pulley.

1. Calculate the acceleration of \(P\) while it is in motion and the corresponding tension in the string.
2. Find the value of \(m\).
3. Calculate the greatest height of \(Q\) above the ground.
4. It is given that the mass of the pulley is 0.5 kg . State the magnitude of the tension in the chain which supports the pulley
   1. when \(P\) is in motion,
   2. when \(P\) is at rest on the ground and \(Q\) is moving upwards.

6 Particle \(P\) of mass 0.3 kg and particle \(Q\) of mass 0.2 kg are 3.6 m apart on a smooth horizontal surface. \(P\) and \(Q\) are simultaneously projected directly towards each other along a straight line. Before the particles collide \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that the particles coalesce in the collision, calculate their common speed after they collide.
2. It is given instead that one particle is at rest immediately after the collision.
   1. State which particle is in motion after the collision and find the speed of this particle.
   2. Find the time taken after the collision for the moving particle to return to its initial position.
   3. On a single diagram sketch the \(( t , v )\) graphs for the two particles, with \(t = 0\) as the instant of their initial projection. \(7 \quad A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(45 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected from \(A\) towards \(B\) with speed \(5 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between the plane and \(P\) is 0.2 .
      1. Given that the level of \(A\) is above the level of \(B\), calculate the speed of \(P\) when it passes through the point \(B\), and the time taken to travel from \(A\) to \(B\).
      2. Given instead that the level of \(A\) is below the level of \(B\),
         (a) show that \(P\) does not reach \(B\),
         (b) calculate the difference in the momentum of \(P\) for the two occasions when it is at \(A\).