CAIE M1 (Mechanics 1) 2015 June

1 A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and the block moves at a constant speed of \(2.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done by the tension in 40 s and find the power applied by the tension.

2
\includegraphics[max width=\textwidth, alt={}, center]{543cb1dd-40e8-4d66-8ca0-4f183f83f366-2_438_903_488_623} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h \mathrm {~m}\) above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(h\).

3 A car of mass 860 kg travels along a straight horizontal road. The power provided by the car's engine is \(P\) W and the resistance to the car's motion is \(R \mathrm {~N}\). The car passes through one point with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The car passes through another point with speed \(22.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

4 A lorry of mass 12000 kg moves up a straight hill of length 500 m , starting at the bottom with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaching the top with a speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 25 m above the level of the bottom of the hill. The resistance to motion of the lorry is 7500 N . Find the driving force of the lorry. \begin{figure}[h] \end{figure} Four coplanar forces of magnitudes \(4 \mathrm {~N} , 8 \mathrm {~N} , 12 \mathrm {~N}\) and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.

1. Find the magnitude and direction of the resultant of the four forces. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{543cb1dd-40e8-4d66-8ca0-4f183f83f366-3_351_629_1260_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).
2. State the magnitude and direction of the resultant of the four forces in Fig. 2.

6 A small box of mass 5 kg is pulled at a constant speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a rough plane inclined at \(10 ^ { \circ }\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

1. Find the coefficient of friction between the box and the plane. The pulling force is removed while the box is moving at \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance moved by the box after the instant at which the pulling force is removed.
   [0pt] [Question 7 is printed on the next page.]

7 A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O 100 \mathrm {~s}\) later. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where $$v = 0.0001 t ^ { 3 } - 0.015 t ^ { 2 } + 0.5 t$$

1. Show that \(P\) is instantaneously at rest when \(t = 0 , t = 50\) and \(t = 100\).
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocitytime graph for \(P\) 's motion for \(0 \leqslant t \leqslant 100\).
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leqslant t \leqslant 100\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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