OCR M1 (Mechanics 1) 2007 January

1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
2. Given also that the driving force of the car is 2100 N , find the resistance to motion of the car.

2
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_583_785_676_680} Three horizontal forces of magnitudes \(15 \mathrm {~N} , 11 \mathrm {~N}\) and 13 N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero.
2. Find the magnitude of the resultant of the three forces.
3. State the direction of the resultant of the three forces.
   \includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_348_711_1804_717} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).
4. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table.
5. A horizontal force of magnitude \(X \mathrm {~N}\), acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\).

4
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-3_216_1146_269_502} Three uniform spheres \(L , M\) and \(N\) have masses \(0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}\) and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(L\) after the collision.
2. \(M\) then collides with \(N\).
   (a) Find the total momentum of \(M\) and \(N\) in the direction of \(M\) 's motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision.
   (b) Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision.

5 A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leqslant t \leqslant 4\) the acceleration is \(1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and for \(4 \leqslant t \leqslant 7\) the particle has constant acceleration \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leqslant t \leqslant 4\).
2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\).
3. Find the displacement of the particle from \(A\) when \(t = 7\).

6
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-4_556_1373_269_386} The diagram shows the ( \(t , v\) ) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s , coming to rest 8 m above ground level.

1. Find the greatest speed reached by the hoist during this stage. The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at \(0.02 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 40 s of the third stage, reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
2. the value of \(V\),
3. the length of time during the third stage for which the hoist is moving at constant speed,
4. the deceleration of the hoist in the final part of the third stage.

7 A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6 .

1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N , correct to 3 significant figures.
2. Find the acceleration of \(P\) when it is moving
   (a) up the plane,
   (b) down the plane.
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   (a) Find the length of time before \(P\) reaches its highest point.
   (b) Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).