3 A circular disc of radius \(a \mathrm {~m}\) has mass per unit area \(\rho \mathrm { kg } \mathrm { m } ^ { - 2 }\) given by \(\rho = k ( a + r )\), where \(r \mathrm {~m}\) is the distance from the centre and \(k\) is a positive constant. The disc can rotate freely about an axis perpendicular to it and through its centre.

1. Show that the mass, \(M \mathrm {~kg}\), of the disc is given by \(M = \frac { 5 } { 3 } k \pi a ^ { 3 }\), and show that the moment of inertia, \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), about this axis is given by \(I = \frac { 27 } { 50 } M a ^ { 2 }\). For the rest of this question, take \(M = 64\) and \(a = 0.625\).
   The disc is at rest when it is given a tangential impulsive blow of 50 N s at a point on its circumference.
2. Find the angular speed of the disc. The disc is then accelerated by a constant couple reaching an angular speed of \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in 20 seconds.
3. Calculate the magnitude of this couple. When the angular speed is \(30 \mathrm { rads } ^ { - 1 }\), the couple is removed and brakes are applied to bring the disc to rest. The effect of the brakes is modelled by a resistive couple of \(3 \dot { \theta } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the disc in \(\mathrm { rad } \mathrm { s } ^ { - 1 }\).
4. Formulate a differential equation for \(\dot { \theta }\) and hence find \(\dot { \theta }\) in terms of \(t\), the time in seconds from when the brakes are first applied.
5. By reference to your expression for \(\dot { \theta }\), give a brief criticism of this model for the effect of the brakes.