CAIE M1 (Mechanics 1) 2008 June

1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the particle,
2. the value of \(\alpha\).

2 A block is being pulled along a horizontal floor by a rope inclined at \(20 ^ { \circ }\) to the horizontal. The tension in the rope is 851 N and the block moves at a constant speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the work done on the block in 12 s is approximately 24 kJ .
2. Hence find the power being applied to the block, giving your answer to the nearest kW .

3
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-2_520_565_1009_792} Three horizontal forces of magnitudes \(F \mathrm {~N} , 13 \mathrm {~N}\) and 10 N act at a fixed point \(O\) and are in equilibrium. The directions of the forces are as shown in the diagram. Find, in either order, the value of \(\theta\) and the value of \(F\).

4
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552}
\(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find

1. the kinetic energy of \(P\) at \(A\),
2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).

5
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_314_867_1457_639} A block \(B\) of mass 0.6 kg and a particle \(A\) of mass 0.4 kg are attached to opposite ends of a light inextensible string. The block is held at rest on a rough horizontal table, and the coefficient of friction between the block and the table is 0.5 . The string passes over a small smooth pulley \(C\) at the edge of the table and \(A\) hangs in equilibrium vertically below \(C\). The part of the string between \(B\) and \(C\) is horizontal and the distance \(B C\) is 3 m (see diagram). \(B\) is released and the system starts to move.

1. Find the acceleration of \(B\) and the tension in the string.
2. Find the time taken for \(B\) to reach the pulley.

6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find

1. the greatest height above the ground reached by \(P\),
2. the speed with which \(P\) reaches the ground,
3. the total work done on \(P\) by the air resistance.

7
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find

1. the maximum velocity of \(P\),
2. the distance \(A B\). \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. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }