5 The region inside the circle \(x ^ { 2 } + y ^ { 2 } = a ^ { 2 }\) is rotated about the \(x\)-axis to form a uniform solid sphere of radius \(a\) and volume \(\frac { 4 } { 3 } \pi a ^ { 3 }\). The mass of the sphere is \(10 M\).

1. Show by integration that the moment of inertia of the sphere about the \(x\)-axis is \(4 M a ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis.) The sphere is free to rotate about a fixed horizontal axis which is a diameter of the sphere. A particle of mass \(M\) is attached to the lowest point of the sphere. The sphere with the particle attached then makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these oscillations.