CAIE Further Paper 3 (Further Paper 3) 2022 November

1 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(O P\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 2 } { 3 }\). The particle \(P\) is projected perpendicular to \(O P\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-03_671_455_255_845} A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(\mathrm { AB } = 9 \mathrm { a }\) and \(\mathrm { BC } = 6 \mathrm { a }\). The point \(D\) is on \(B C\) such that \(\mathrm { BD } = \mathrm { x }\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.

3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 16 } { 3 } \mathrm { Mg }\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4 M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2 M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac { 1 } { 4 } a\) is \(v\). Find an expression for \(v\) in terms of \(g\) and \(a\).

4 A particle \(P\) of mass 5 kg moves along a horizontal straight line. At time \(t \mathrm {~s}\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from a fixed point \(O\) on the line is \(x \mathrm {~m}\). The forces acting on \(P\) are a force of magnitude \(\frac { 500 } { v } \mathrm {~N}\) in the direction \(O P\) and a resistive force of magnitude \(\frac { 1 } { 2 } v ^ { 2 } \mathrm {~N}\). When \(t = 0 , x = 0\) and \(v = 5\).

1. Find an expression for \(v\) in terms of \(x\).
2. State the value that the speed approaches for large values of \(x\).

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t \mathrm {~s}\) are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Show that the equation of the trajectory is given by $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ In the subsequent motion \(P\) passes through the point with coordinates \(( 30,20 )\).
2. Given that one possible value of \(\tan \theta\) is \(\frac { 4 } { 3 }\), find the other possible value of \(\tan \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-10_451_1339_258_404} A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. One end of the string is attached to a particle \(A\) of mass \(m\). The other end of the string is attached to a particle \(B\) of mass \(\frac { 6 } { 7 } m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram). When \(A\) has constant angular speed \(\omega\), the angle between \(A R\) and \(B R\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\). When \(A\) has constant angular speed \(\frac { 3 } { 2 } \omega\), the angle between \(A R\) and \(B R\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac { 1 } { 2 } N\).
3. Show that \(\cos \theta = \frac { 4 } { 9 } \cos \alpha\).
4. Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\).
   \includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-12_413_974_255_587} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 1 } { 2 } m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2 u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 5 } { 8 }\) and \(\alpha + \beta = 90 ^ { \circ }\).
5. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\).
   The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
6. Find the value of \(\tan \alpha\).
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