OCR M1 (Mechanics 1) 2011 June

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.

2 Particles \(P\) and \(Q\), of masses 0.45 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley. The particles are released from rest with the string taut and both particles 0.36 m above a horizontal surface. \(Q\) descends with acceleration \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). When \(Q\) strikes the surface, it remains at rest.

1. Calculate the tension in the string while both particles are in motion.
2. Find the value of \(m\).
3. Calculate the speed at which \(Q\) strikes the surface.
4. Calculate the greatest height of \(P\) above the surface. (You may assume that \(P\) does not reach the pulley.)

3 A block \(B\) of mass 0.8 kg is pulled across a horizontal surface by a force of 6 N inclined at an angle of \(60 ^ { \circ }\) to the upward vertical. The coefficient of friction between the block and the surface is 0.2 . Calculate

1. the vertical component of the force exerted on \(B\) by the surface,
2. the acceleration of \(B\). The 6 N force is removed when \(B\) has speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Calculate the time taken for \(B\) to decelerate from a speed of \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to rest.

4
\includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-2_657_1495_1539_324} A car travelling on a straight road accelerates from rest to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . It continues at constant speed for 11 s and then decelerates to rest in 2 s . The driver gets out of the car and walks at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 20 s back to a shop which he enters. Some time later he leaves the shop and jogs to the car at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He arrives at the vehicle 60 s after it began to accelerate from rest. The diagram, which has six straight line segments, shows the \(( t , v )\) graph for the motion of the driver.

1. Calculate the initial acceleration and final deceleration of the car.
2. Calculate the distance the car travels.
3. Calculate the length of time the driver is in the shop.

5
\includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_362_1065_258_539} Three particles \(P , Q\) and \(R\) lie on a line of greatest slope of a smooth inclined plane. \(P\) has mass 0.5 kg and initially is at the foot of the plane. \(R\) has mass 0.3 kg and initially is at the top of the plane. \(Q\) has mass 0.2 kg and is between \(P\) and \(R\) (see diagram). \(P\) is projected up the line of greatest slope with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant when \(Q\) and \(R\) are released from rest. Each particle has an acceleration of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) down the plane.

1. \(P\) and \(Q\) collide 0.4 s after being set in motion. Immediately after the collision \(Q\) moves up the plane with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed and direction of motion of \(P\) immediately after the collision.
2. 0.6 s after its collision with \(P , Q\) collides with \(R\) and the two particles coalesce. Find the speed and direction of motion of the combined particle immediately after the collision

6
\includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_348_1109_1345_516} A small smooth ring \(R\) of weight 7 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\) at the same horizontal level. A horizontal force of magnitude 5 N is applied to \(R\). The string is taut. In the equilibrium position the angle \(A R B\) is a right angle, and the portion of the string attached to \(B\) makes an angle \(\theta\) with the horizontal (see diagram).

1. Explain why the tension \(T \mathrm {~N}\) is the same in each part of the string.
2. By resolving horizontally and vertically for the forces acting on \(R\), form two simultaneous equations in \(T \cos \theta\) and \(T \sin \theta\).
3. Hence find \(T\) and \(\theta\).

7 A particle \(P\) is projected from a fixed point \(O\) on a straight line. The displacement \(x\) m of \(P\) from \(O\) at time \(t \mathrm {~s}\) after projection is given by \(x = 0.1 t ^ { 3 } - 0.3 t ^ { 2 } + 0.2 t\).

1. Express the velocity and acceleration of \(P\) in terms of \(t\).
2. Show that when the acceleration of \(P\) is zero, \(P\) is at \(O\).
3. Find the values of \(t\) when \(P\) is stationary. At the instant when \(P\) first leaves \(O\), a particle \(Q\) is projected from \(O\). \(Q\) moves on the same straight line as \(P\) and at time \(t \mathrm {~s}\) after projection the velocity of \(Q\) is given by \(\left( 0.2 t ^ { 2 } - 0.4 \right) \mathrm { ms } ^ { - 1 } . P\) and \(Q\) collide first when \(t = T\).
4. Show that \(T\) satisfies the equation \(t ^ { 2 } - 9 t + 18 = 0\), and hence find \(T\).