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.\\
(i) 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.\\
(ii) 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.\\
(iii) 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 $\mathrm { rad } \mathrm { s } ^ { - 1 }$.\\
(iv) 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.\\
(v) By reference to your expression for $\dot { \theta }$, give a brief criticism of this model for the effect of the brakes.

\hfill \mbox{\textit{OCR MEI M4 2008 Q3 [24]}}