CAIE FP2 (Further Pure Mathematics 2) 2010 June

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\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-2_159_707_1443_721} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.

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\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-3_506_969_255_587} Two coplanar discs, of radii 0.5 m and 0.3 m , rotate about their centres \(A\) and \(B\) respectively, where \(A B = 0.8 \mathrm {~m}\). At time \(t\) seconds the angular speed of the larger disc is \(\frac { 1 } { 2 } t \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find

1. the angular speed of the smaller disc,
2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(P A\).

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\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-3_378_625_1272_758} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m , g\) and \(\beta\), and hence show that \(\beta = \frac { 1 } { 4 } \pi\). The particle is given a velocity of magnitude \(\sqrt { } ( a g )\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.

A uniform disc, of mass \(4 m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2 a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(O A , O B\) and \(O C\), each of mass \(m\) and length \(2 a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42 m a ^ { 2 }\). The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(A O\) making an angle of \(30 ^ { \circ }\) with the horizontal. Find the angular speed of the wheel when \(A O\) is horizontal. When \(A O\) is horizontal the disc becomes detached from the wheel. Find the angle that \(A O\) makes with the horizontal when the wheel first comes to instantaneous rest.