CAIE M1 (Mechanics 1) 2012 June

1 A car of mass 880 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to motion is 700 N . At an instant when the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(P\).

2
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_318_632_482_753} Forces of magnitudes 13 N and 14 N act at a point \(O\) in the directions shown in the diagram. The resultant of these forces has magnitude 15 N . Find

1. the value of \(\theta\),
2. the component of the resultant in the direction of the force of magnitude 14 N .

3
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_502_661_1219_742} A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point \(O\) on the ground, using a winding drum. The load passes through a point \(A , 20 \mathrm {~m}\) above \(O\), with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Find, for the motion from \(O\) to \(A\),

1. the gain in the potential energy of the load,
2. the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.
3. Given that the work done against the resistance to motion from \(O\) to \(A\) is 20 kJ and that the time taken for the load to travel from \(O\) to \(A\) is 41.7 s , find the power output of the winding drum.

4 A particle \(P\) starts at the point \(O\) and travels in a straight line. At time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.75 t ^ { 2 } - 0.0625 t ^ { 3 }\). Find

1. the positive value of \(t\) for which the acceleration is zero,
2. the distance travelled by \(P\) before it changes its direction of motion.

5
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_485_874_255_638} The diagram shows the vertical cross-section \(O A B\) of a slide. The straight line \(A B\) is tangential to the curve \(O A\) at \(A\). The line \(A B\) is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The point \(O\) is 10 m higher than \(B\), and \(A B\) has length 10 m (see diagram). The part of the slide containing the curve \(O A\) is smooth and the part containing \(A B\) is rough. A particle \(P\) of mass 2 kg is released from rest at \(O\) and moves down the slide.

1. Find the speed of \(P\) when it passes through \(A\). The coefficient of friction between \(P\) and the part of the slide containing \(A B\) is \(\frac { 1 } { 12 }\). Find
2. the acceleration of \(P\) when it is moving from \(A\) to \(B\),
3. the speed of \(P\) when it reaches \(B\).

6
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_465_849_1475_648} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle \(\theta\) with the ground, where \(\sin \theta = 0.8\). Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.

1. Find the tension in the string and the acceleration of the particles while both are moving. The speed of \(P\) when it reaches the ground is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On reaching the ground \(P\) comes to rest and remains at rest. \(Q\) continues to move up the slope but does not reach the pulley.
2. Find the time taken from the instant that the particles are released until \(Q\) reaches its greatest height above the ground.

7
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-4_529_481_255_831} A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end \(A\) of a light inextensible string is attached to the ring. The other end \(C\) of the string is attached to a fixed point of the rod above \(A\). A horizontal force of magnitude 8 N is applied to the point \(B\) of the string, where \(A B = 1.5 \mathrm {~m}\) and \(B C = 2 \mathrm {~m}\). The system is in equilibrium with the string taut and \(A B\) at right angles to \(B C\) (see diagram).

1. Find the tension in the part \(A B\) of the string and the tension in the part \(B C\) of the string. The equilibrium is limiting with the ring on the point of sliding up the rod.
2. Find the coefficient of friction between the ring and the rod.