CAIE M2 (Mechanics 2) 2013 November

1 A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(O\) of a smooth horizontal plane. \(P\) moves on the plane at constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circle with centre \(O\). Calculate the tension in the string.

2
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-2_359_686_484_731} A uniform frame consists of a semicircular arc \(A B C\) of radius 0.6 m together with its diameter \(A O C\), where \(O\) is the centre of the semicircle (see diagram).

1. Calculate the distance of the centre of mass of the frame from \(O\). The frame is freely suspended at \(A\) and hangs in equilibrium.
2. Calculate the angle between \(A C\) and the vertical.

3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).

4 A small ball \(B\) is projected from a point \(O\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal.

1. Calculate the speed and direction of motion of \(B\) for the instant 1.8 s after projection. The point \(O\) is 2 m above a horizontal plane.
2. Calculate the time after projection when \(B\) reaches the plane.

5
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_540_537_255_804} A particle \(P\) of mass 0.2 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.4 m . A second light inextensible string of length 0.3 m connects \(P\) to a fixed point \(B\) which is vertically below \(A\). The particle \(P\) moves in a horizontal circle, which has its centre on the line \(A B\), with the angle \(A P B = 90 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, calculate the speed of \(P\).
2. It is given instead that \(P\) moves with its least possible angular speed for motion in this circle. Find this angular speed.

6
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_454_1029_1379_557}
\(A B C D\) is the cross-section through the centre of mass of a uniform rectangular block of weight 260 N . The lengths \(A B\) and \(B C\) are 1.5 m and 0.8 m respectively. The block rests in equilibrium with the point \(D\) on a rough horizontal floor. Equilibrium is maintained by a light rope attached to the point \(A\) on the block and the point \(E\) on the floor. The points \(E , A\) and \(B\) lie in a straight line inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).

1. By taking moments about \(D\), show that the tension in the rope is 146 N , correct to 3 significant figures.
2. Given that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the floor.

7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
2. Calculate the speed of \(P\) immediately before it collides with the plane.
3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.