CAIE M2 (Mechanics 2) 2008 June

1
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_549_775_269_685} A particle \(A\) and a block \(B\) are attached to opposite ends of a light elastic string of natural length 2 m and modulus of elasticity 6 N . The block is at rest on a rough horizontal table. The string passes over a small smooth pulley \(P\) at the edge of the table, with the part \(B P\) of the string horizontal and of length 1.2 m . The frictional force acting on \(B\) is 1.5 N and the system is in equilibrium (see diagram). Find the distance \(P A\).

2
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_686_495_1238_826} A uniform rigid wire \(A B\) is in the form of a circular arc of radius 1.5 m with centre \(O\). The angle \(A O B\) is a right angle. The wire is in equilibrium, freely suspended from the end \(A\). The chord \(A B\) makes an angle of \(\theta ^ { \circ }\) with the vertical (see diagram).

1. Show that the distance of the centre of mass of the arc from \(O\) is 1.35 m , correct to 3 significant figures.
2. Find the value of \(\theta\).

3
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

4
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_375_627_1448_758} Uniform rods \(A B , A C\) and \(B C\) have lengths \(3 \mathrm {~m} , 4 \mathrm {~m}\) and 5 m respectively, and weights \(15 \mathrm {~N} , 20 \mathrm {~N}\) and 25 N respectively. The rods are rigidly joined to form a right-angled triangular frame \(A B C\). The frame is hinged at \(B\) to a fixed point and is held in equilibrium, with \(A C\) horizontal, by means of an inextensible string attached at \(C\). The string is at right angles to \(B C\) and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Find the value of \(T\). A uniform triangular lamina \(P Q R\), of weight 60 N , has the same size and shape as the frame \(A B C\). The lamina is now attached to the frame with \(P , Q\) and \(R\) at \(A , B\) and \(C\) respectively. The composite body is held in equilibrium with \(A , B\) and \(C\) in the same positions as before. Find
2. the new value of \(T\),
3. the magnitude of the vertical component of the force acting on the composite body at \(B\).

5
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-4_547_933_269_607} Particles \(A\) and \(B\) are projected simultaneously from the top \(T\) of a vertical tower, and move in the same vertical plane. \(T\) is 7.2 m above horizontal ground. \(A\) is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is projected at an angle of \(60 ^ { \circ }\) above the horizontal with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } . A\) and \(B\) move away from each other (see diagram).

1. Find the time taken for \(A\) to reach the ground. At the instant when \(A\) hits the ground,
2. show that \(B\) is approximately 5.2 m above the ground,
3. find the distance \(A B\).

6 One end of a light elastic string of natural length 1.25 m and modulus of elasticity 20 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.5 kg is attached to the other end of the string. \(P\) is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = - 32 x ^ { 2 } + 20 x + 25\).
2. Find the maximum speed of \(P\).
3. Find the acceleration of \(P\) when it is at its lowest point.

7 A particle \(P\) of mass 0.5 kg moves on a horizontal surface along the straight line \(O A\), in the direction from \(O\) to \(A\). The coefficient of friction between \(P\) and the surface is 0.08 . Air resistance of magnitude \(0.2 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\). The particle passes through \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0\).

1. Show that \(2.5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 2 )\) and hence find the value of \(t\) when \(v = 0\).
2. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 6 \mathrm { e } ^ { - 0.4 t } - 2\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance \(O P\) when \(v = 0\). \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. }