CAIE FP2 (Further Pure Mathematics 2) 2013 June

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\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-2_684_714_246_717} A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(4 a\), rests with the end \(A\) on rough horizontal ground. The point \(C\) on \(A B\) is such that \(A C = 3 a\). A light inextensible string has one end attached to the point \(P\) which is at a distance \(5 a\) vertically above \(A\), and the other end attached to \(C\). The rod and the string are in the same vertical plane and the system is in equilibrium with angle \(A C P\) equal to \(90 ^ { \circ }\) (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). Show that the least possible value of \(\mu\) is \(\frac { 24 } { 43 }\).

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\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-3_906_1538_248_301} The end \(A\) of a uniform \(\operatorname { rod } A B\), of mass \(4 m\) and length \(3 a\), is rigidly attached to a point on a uniform spherical shell, of mass \(\lambda m\) and radius \(3 a\). The end \(B\) of the rod is rigidly attached to a point on a uniform ring. The ring has centre \(O\), mass \(4 m\) and radius \(\frac { 1 } { 2 } a\). The ring and the rod are in the same vertical plane. The line \(O B A\), extended, passes through the centre of the spherical shell. \(B C\) is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through \(C\) perpendicular to the plane of the ring, is \(( 30 + 55 \lambda ) m a ^ { 2 }\). Given that the system performs small oscillations of period \(2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)\) about this axis, find the value of \(\lambda\).