CAIE M2 (Mechanics 2) 2016 June

2
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-2_515_463_484_842} A uniform wire has the shape of a semicircular arc, with diameter \(A B\) of length 0.8 m . The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(A B\) inclined at \(70 ^ { \circ }\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall 0.8 m vertically above \(A\). The tension in the string is 15 N (see diagram).

1. Show that the horizontal distance of the centre of mass of the wire from the wall is 0.463 m , correct to 3 significant figures.
2. Calculate the weight of the wire.

3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) down the plane, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 \mathrm { e } ^ { - x } \mathrm {~N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 2 \mathrm { e } ^ { - x }\).
2. Find \(v\) when \(x = 0.6\).
   \includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-3_905_604_251_769} A uniform solid cone has base radius 0.4 m and height 4.4 m . A uniform solid cylinder has radius 0.4 m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal (see diagram).
3. Calculate the least possible value of the coefficient of friction between the plane and the object.
4. Calculate the greatest possible height of the cylinder.

6
\includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-4_503_805_260_671} A light inextensible string passes through a small smooth bead \(B\) of mass 0.4 kg . One end of the string is attached to a fixed point \(A 0.4 \mathrm {~m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius 0.3 m (see diagram).

1. Given that the tension in the string is 2 N , calculate
   (a) the angular speed of the bead,
   (b) the magnitude of the contact force exerted on the bead by the surface.
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.