CAIE M1 (Mechanics 1) 2003 November

1 A motorcycle of mass 100 kg is travelling on a horizontal straight road. Its engine is working at a rate of 8 kW . At an instant when the speed of the motorcycle is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find, at this instant,

1. the force produced by the engine,
2. the resistance to motion of the motorcycle.

2 A stone is released from rest and falls freely under gravity. Find

1. the speed of the stone after 2 s ,
2. the time taken for the stone to fall a distance of 45 m from its initial position,
3. the distance fallen by the stone from the instant when its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to the instant when its speed is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_143_611_1050_769} A crate of mass 3 kg is pulled at constant speed along a horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(15 ^ { \circ }\) to the horizontal, as shown in the diagram. Find

1. the work done by the pulling force in moving the crate a distance of 2 m ,
2. the normal component of the contact force on the crate.

4
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_227_586_1631_781} The diagram shows a vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section. A particle of mass 0.15 kg is released from rest at \(A\).

1. Assuming that the particle reaches \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there are no resistances to motion, find the height of \(A\) above \(B\).
2. Assuming instead that the particle reaches \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and that the height of \(A\) above \(B\) is 4 m , find the work done against the resistances to motion.

5
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_300_792_274_680} Particles \(A\) and \(B\), of masses 0.4 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a horizontal table with the string passing over a smooth pulley at the edge of the table. Particle \(B\) hangs vertically below the pulley (see diagram). The system is released from rest. In the subsequent motion a constant frictional force of magnitude 0.6 N acts on \(A\). Find

1. the tension in the string,
2. the speed of \(B 1.5 \mathrm {~s}\) after it starts to move.

6
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_579_469_1142_840} One end of a light inextensible string is attached to a fixed point \(A\) of a fixed vertical wire. The other end of the string is attached to a small ring \(B\), of mass 0.2 kg , through which the wire passes. A horizontal force of magnitude 5 N is applied to the mid-point \(M\) of the string. The system is in equilibrium with the string taut, with \(B\) below \(A\), and with angles \(A B M\) and \(B A M\) equal to \(30 ^ { \circ }\) (see diagram).

1. Show that the tension in \(B M\) is 5 N .
2. The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.
3. A particle of mass \(m \mathrm {~kg}\) is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of \(m\).

7
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-4_547_1237_269_456} A tractor \(A\) starts from rest and travels along a straight road for 500 seconds. The velocity-time graph for the journey is shown above. This graph consists of three straight line segments. Find

1. the distance travelled by \(A\),
2. the initial acceleration of \(A\). Another tractor \(B\) starts from rest at the same instant as \(A\), and travels along the same road for 500 seconds. Its velocity \(t\) seconds after starting is \(\left( 0.06 t - 0.00012 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
3. how much greater \(B\) 's initial acceleration is than \(A\) 's,
4. how much further \(B\) has travelled than \(A\), at the instant when \(B\) 's velocity reaches its maximum.