CAIE FP2 (Further Pure Mathematics 2) 2012 June

3 A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { } \left( \frac { 7 } { 2 } g a \right)\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(O P\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac { 3 } { 2 } m g ( 1 + 2 \cos \theta )\). Find the speed of \(P\)

1. when it loses contact with the sphere,
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.)
   \(4 A B\) is a diameter of a uniform circular disc \(D\) of mass \(9 m\), radius \(3 a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112 m a ^ { 2 }\). A particle of mass \(3 m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k \sqrt { } ( g a )\). Find the value of \(k\), correct to 3 significant figures.

5
\includegraphics[max width=\textwidth, alt={}, center]{621b50d6-44e8-435d-ac6e-bb2ee5bcdd38-2_478_867_1816_641} Two uniform rods \(A B\) and \(B C\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The \(\operatorname { rod } A B\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(A B\) has mass \(3 m\) and length \(3 a\), the rod \(B C\) has mass \(5 m\) and length \(5 a\), and \(C\) is at a distance \(6 a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(B C\) at \(C\) has magnitude \(\frac { 13 } { 2 } m g\). The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\).

8 The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.016 x } & x \geqslant 0
0 & x < 0 \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. Find

1. the median distance between two successive flaws,
2. the probability that there is a distance of at least 50 metres between two successive flaws.