3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(4 m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below \(A\).

1. Show that \(\mathrm { k } = \frac { 4 \mathrm { a } } { \mathrm { x } - \mathrm { a } }\).
   An additional particle, of mass \(2 m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac { 1 } { 3 } a\), its speed is \(\frac { 1 } { 3 } \sqrt { \mathrm { ga } }\).
2. Find \(x\) in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-06_602_520_264_753} A uniform solid circular cone has vertical height \(k h\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
3. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac { \mathrm { h } \left( \mathrm { k } ^ { 2 } + 4 \mathrm { k } + 6 \right) } { 4 ( 3 + \mathrm { k } ) }\).
   The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3 h = 2 r\) and that the solid is on the point of toppling when \(\tan \theta = \frac { 4 } { 3 }\).
4. Find the value of \(k\).