CAIE M1 (Mechanics 1) 2015 November

1
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-2_558_529_258_808} Four horizontal forces act at a point \(O\) and are in equilibrium. The magnitudes of the forces are \(F \mathrm {~N}\), \(G \mathrm {~N} , 15 \mathrm {~N}\) and \(F \mathrm {~N}\), and the forces act in directions as shown in the diagram.

1. Show that \(F = 41.0\), correct to 3 significant figures.
2. Find the value of \(G\).

2 A particle is released from rest at a point \(H \mathrm {~m}\) above horizontal ground and falls vertically. The particle passes through a point 35 m above the ground with a speed of \(( V - 10 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and reaches the ground with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(V\),
2. the value of \(H\).

3 A particle \(P\) moves along a straight line for 100 s . It starts at a point \(O\) and at time \(t\) seconds after leaving \(O\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.00004 t ^ { 3 } - 0.006 t ^ { 2 } + 0.288 t$$

1. Find the values of \(t\) at which the acceleration of \(P\) is zero.
2. Find the displacement of \(P\) from \(O\) when \(t = 100\).

4
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).

5
\includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.

1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the length of the string.

6 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small ring of mass 0.024 kg is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude 0.195 N at an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 5 } { 13 }\). When the angle \(\theta\) is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.

1. Find the coefficient of friction between the ring and the rod. When the angle \(\theta\) is above the horizontal (see Fig. 2) the ring moves.
2. Find the acceleration of the ring.

7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.

1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
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