CAIE M1 (Mechanics 1) 2015 June

1 One end of a light inextensible string is attached to a block. The string makes an angle of \(60 ^ { \circ }\) above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the string is 8 N . The block starts to move with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 5 s of the block's motion, find

1. the distance travelled,
2. the work done by the tension in the string.

2 The total mass of a cyclist and his cycle is 80 kg . The resistance to motion is zero.

1. The cyclist moves along a horizontal straight road working at a constant rate of \(P \mathrm {~W}\). Find the value of \(P\) given that the cyclist's speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when his acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The cyclist moves up a straight hill inclined at an angle \(\alpha\), where \(\sin \alpha = 0.035\). Find the acceleration of the cyclist at an instant when he is working at a rate of 450 W and has speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A plane is inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\) to the horizontal. \(A\) and \(B\) are two points on the same line of greatest slope with \(A\) higher than \(B\). The distance \(A B\) is 12 m . A small object \(P\) of mass 8 kg is released from rest at \(A\) and slides down the plane, passing through \(B\) with speed \(4.5 \mathrm {~ms} ^ { - 1 }\). For the motion of \(P\) from \(A\) to \(B\), find

1. the increase in kinetic energy of \(P\) and the decrease in potential energy of \(P\),
2. the magnitude of the constant resisting force that opposes the motion of \(P\).

4 A particle \(P\) moves in a straight line. At time \(t\) seconds after starting from rest at the point \(O\) on the line, the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.075 t ^ { 2 } - 1.5 t + 5\).

1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
2. Hence find the time taken for \(P\) to return to the point \(O\).

5 A particle \(P\) starts from rest at a point \(O\) on a horizontal straight line. \(P\) moves along the line with constant acceleration and reaches a point \(A\) on the line with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the instant that \(P\) leaves \(O\), a particle \(Q\) is projected vertically upwards from the point \(A\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(P\) and \(Q\) collide at \(A\). Find

1. the acceleration of \(P\),
2. the distance \(O A\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_538_414_315_370} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_561_686_264_1080} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Two particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively. The particles are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest with the string taut and both straight parts of the string vertical. \(P\) and \(Q\) are each at a height of \(h \mathrm {~m}\) above horizontal ground (see Fig. 1). \(P\) is released and \(Q\) moves downwards. Subsequently \(Q\) hits the ground and comes to rest. Fig. 2 shows the velocity-time graph for \(P\) while \(Q\) is moving downwards or is at rest on the ground.

1. Find the value of \(h\).
2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.

7
\includegraphics[max width=\textwidth, alt={}, center]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-4_657_618_255_760} A small ring \(R\) is attached to one end of a light inextensible string of length 70 cm . A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point \(A\) on the wire, vertically above \(R\). A horizontal force of magnitude 5.6 N is applied to the point \(J\) of the string 30 cm from \(A\) and 40 cm from \(R\). The system is in equilibrium with each of the parts \(A J\) and \(J R\) of the string taut and angle \(A J R\) equal to \(90 ^ { \circ }\) (see diagram).

1. Find the tension in the part \(A J\) of the string, and find the tension in the part \(J R\) of the string. The ring \(R\) has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
2. Show that the coefficient of friction between \(R\) and the wire is 0.341 , correct to 3 significant figures. A particle of mass \(m \mathrm {~kg}\) is attached to \(R\) and \(R\) is now in limiting equilibrium, on the point of moving down the wire.
3. Given that the coefficient of friction is unchanged, find the value of \(m\). \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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. }