4 In this question you may assume without proof the standard results in Examination Formulae and Tables (MF2) for

• the moment of inertia of a disc about an axis through its centre perpendicular to the disc,
• the position of the centre of mass of a solid uniform cone.

Fig. 4 shows a uniform cone of radius \(a\) and height \(2 a\), with its axis of symmetry on the \(x\)-axis and its vertex at the origin. A thin slice through the cone parallel to the base is at a distance \(x\) from the vertex. \begin{figure}[h] \end{figure} The slice is taken to be a thin uniform disc of mass \(m\).

1. Write down the moment of inertia of the disc about the \(x\)-axis. Hence show that the moment of inertia of the disc about the \(y\)-axis is \(\frac { 17 } { 16 } m x ^ { 2 }\).
2. Hence show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 51 } { 20 } M a ^ { 2 }\), where \(M\) is the mass of the cone. [You may assume without proof the formula for the volume of a cone.] The cone is now suspended so that it can rotate freely about a fixed, horizontal axis through its vertex. The axis of symmetry of the cone moves in a vertical plane perpendicular to the axis of rotation. The cone is released from rest when its axis of symmetry is at an acute angle \(\alpha\) to the downward vertical. At time \(t\), the angle the axis of symmetry makes with the downward vertical is \(\theta\).
3. Use an energy method to show that \(\dot { \theta } ^ { 2 } = \frac { 20 g } { 17 a } ( \cos \theta - \cos \alpha )\).
4. Hence, or otherwise, show that if \(\alpha\) is small the cone performs approximate simple harmonic motion and find the period. RECOGNISING ACHIEVEMENT