CAIE M1 (Mechanics 1) 2007 November

1 A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant rate of \(P \mathrm {~kW}\). The resistance to motion of the car is 550 N . Given that the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(P\).

2 A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet.

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\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-2_429_826_721_662} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.

1. Find the values of \(F\) and \(\theta\).
2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces.

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\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-2_560_591_1663_776} The diagram shows the vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section, and \(A\) is 5 m higher than \(B\). A particle of mass 0.35 kg passes through \(A\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), moving on the surface towards \(B\).

1. Assuming that there is no resistance to motion, find the speed with which the particle reaches \(B\).
2. Assuming instead that there is a resistance to motion, and that the particle reaches \(B\) with speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against this resistance as the particle moves from \(A\) to \(B\). A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T \mathrm {~N}\) acting at an angle of \(60 ^ { \circ }\) to the downward vertical (see diagram). The ring is in equilibrium.

1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R \mathrm {~N}\) and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\).
2. The coefficient of friction between the rod and the ring is 0.7 . Find the value of \(T\) for which the ring is about to slip.
3. A man walks in a straight line from \(A\) to \(B\) with constant acceleration \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his speed at \(B\) is \(2.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the time taken for the man to walk from \(A\) to \(B\), and find the distance \(A B\).
4. A woman cyclist leaves \(A\) at the same instant as the man. She starts from rest and travels in a straight line to \(B\), reaching \(B\) at the same instant as the man. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k \left( 200 t - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find
   (a) the value of \(k\),
   (b) the cyclist's speed at \(B\).
5. Sketch, using the same axes, the velocity-time graphs for the man's motion and the woman's motion from \(A\) to \(B\).

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\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-4_309_1209_269_468} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses 0.13 kg and 0.11 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.

1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane.
2. Show that \(T - F > 0.32\), where \(T \mathrm {~N}\) is the tension in the string and \(F \mathrm {~N}\) is the magnitude of the frictional force on \(P\). The coefficient of friction between \(P\) and the plane is 0.6 .
3. Find the acceleration of \(P\). \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. }