CAIE M1 (Mechanics 1) 2009 June

1
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_388_565_264_790} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground.

2
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_349_1139_1011_502} A crate \(C\) is pulled at constant speed up a straight inclined path by a constant force of magnitude \(F \mathrm {~N}\), acting upwards at an angle of \(15 ^ { \circ }\) to the path. \(C\) passes through points \(P\) and \(Q\) which are 100 m apart (see diagram). As \(C\) travels from \(P\) to \(Q\) the work done against the resistance to \(C\) 's motion is 900 J , and the gain in \(C\) 's potential energy is 2100 J . Write down the work done by the pulling force as \(C\) travels from \(P\) to \(Q\), and hence find the value of \(F\).

3
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_492_606_1763_772} Forces of magnitudes \(7 \mathrm {~N} , 10 \mathrm {~N}\) and 15 N act on a particle in the directions shown in the diagram.

1. Find the component of the resultant of the three forces
   (a) in the \(x\)-direction,
   (b) in the \(y\)-direction.
2. Hence find the direction of the resultant.
   \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_414_833_267_657} A block of mass 8 kg is at rest on a plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).
3. Given that the tension in the string is 13 N , find the frictional and normal components of the force exerted on the block by the plane. The string is cut; the block remains at rest, but is on the point of slipping down the plane.
4. Find the coefficient of friction between the block and the plane.

5
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_165_1417_1320_365} A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest at the top \(A\) of a straight path \(A B\), and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(A B\) is inclined at \(2.6 ^ { \circ }\) to the horizontal and is of length 250 m (see diagram).

1. Given that the cyclist passes through \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. The cyclist continues to freewheel along a horizontal straight path \(B D\) until he reaches the point \(C\), where the distance \(B C\) is \(d \mathrm {~m}\). His speed at \(C\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant, and is the same on \(B D\) as on \(A B\).
2. Find the value of \(d\). The cyclist starts to pedal at \(C\), generating 425 W of power.
3. Find the acceleration of the cyclist immediately after passing through \(C\).

6
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-4_712_526_264_813} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s .

1. Find the acceleration of \(A\) as it falls. The mass of \(A\) is 0.45 kg . Find
2. the tension in the string while \(A\) is falling,
3. the mass of \(B\),
4. the maximum height above the floor reached by \(B\).

7 A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(A B\) the velocity of the particle is \(\left( 0.5 t - 0.01 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(t \mathrm {~s}\) is the time after leaving \(A\).

1. Given that the acceleration of \(P\) at \(B\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the time taken for \(P\) to travel from \(A\) to \(B\). The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that \(P\) reaches \(C\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken for \(P\) to travel from \(B\) to \(C\).
   \(P\) travels with constant deceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(C\) to \(D\). Given that the distance \(C D\) is 300 m , find
3. the speed with which \(P\) reaches \(D\),
4. the distance \(A D\). \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. }