CAIE M1 (Mechanics 1) 2012 June

1 A block is pulled in a straight line along horizontal ground by a force of constant magnitude acting at an angle of \(60 ^ { \circ }\) above the horizontal. The work done by the force in moving the block a distance of 5 m is 75 J . Find the magnitude of the force.

2
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_465_478_479_836} Three coplanar forces of magnitudes \(F \mathrm {~N} , 12 \mathrm {~N}\) and 15 N are in equilibrium acting at a point \(P\) in the directions shown in the diagram. Find \(\alpha\) and \(F\).

3 A particle \(P\) moves in a straight line, starting from the point \(O\) with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(2 t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(t ^ { \frac { 5 } { 3 } } = \frac { 5 } { 6 }\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance of \(P\) from \(O\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_168_711_1612_717} A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of \(25 ^ { \circ }\) below the horizontal (see diagram). With a tension in the string of \(T \mathrm {~N}\) the ring is in equilibrium.

1. Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod. The coefficient of friction between the ring and the rod is 0.4 .
2. Given that the equilibrium is limiting, find the value of \(T\).

5
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_529_195_255_977} A block \(A\) of mass 3 kg is attached to one end of a light inextensible string \(S _ { 1 }\). Another block \(B\) of mass 2 kg is attached to the other end of \(S _ { 1 }\), and is also attached to one end of another light inextensible string \(S _ { 2 }\). The other end of \(S _ { 2 }\) is attached to a fixed point \(O\) and the blocks hang in equilibrium below \(O\) (see diagram).

1. Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\). The string \(S _ { 2 }\) breaks and the particles fall. The air resistance on \(A\) is 1.6 N and the air resistance on \(B\) is 4 N .
2. Find the acceleration of the particles and the tension in \(S _ { 1 }\).

6 A car of mass 1250 kg travels from the bottom to the top of a straight hill which has length 400 m and is inclined to the horizontal at an angle of \(\alpha\), where \(\sin \alpha = 0.125\). The resistance to the car's motion is 800 N . Find the work done by the car's engine in each of the following cases.

1. The car's speed is constant.
2. The car's initial speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car's driving force is 3 times greater at the top of the hill than it is at the bottom, and the car's power output is 5 times greater at the top of the hill than it is at the bottom.

7
\includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_168_803_1909_671} The frictional force acting on a small block of mass 0.15 kg , while it is moving on a horizontal surface, has magnitude 0.12 N . The block is set in motion from a point \(X\) on the surface, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits a vertical wall at a point \(Y\) on the surface 2 s later. The block rebounds from the wall and moves directly towards \(X\) before coming to rest at the point \(Z\) (see diagram). At the instant that the block hits the wall it loses 0.072 J of its kinetic energy. The velocity of the block, in the direction from \(X\) to \(Y\), is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after it leaves \(X\).

1. Find the values of \(v\) when the block arrives at \(Y\) and when it leaves \(Y\), and find also the value of \(t\) when the block comes to rest at \(Z\). Sketch the velocity-time graph.
2. The displacement of the block from \(X\), in the direction from \(X\) to \(Y\), is \(s \mathrm {~m}\) at time \(t \mathrm {~s}\). Sketch the displacement-time graph. Show on your graph the values of \(s\) and \(t\) when the block is at \(Y\) and when it comes to rest at \(Z\).