OCR MEI M4 (Mechanics 4) 2008 June

1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) in terms of \(t\).
3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).

2 A car of mass \(m \mathrm {~kg}\) starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power \(P\) watts, given by \(P = m \left( k ^ { 2 } - v ^ { 2 } \right)\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the car and \(k\) is a positive constant. The displacement from O in the direction of motion is \(x \mathrm {~m}\).

1. Show that \(\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\), and hence find \(x\) in terms of \(v\) and \(k\).
2. How far does the car travel before reaching \(90 \%\) of its terminal velocity?

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.

4 A uniform smooth pulley can rotate freely about its axis, which is fixed and horizontal. A light elastic string AB is attached to the pulley at the end B . The end A is attached to a fixed point such that the string is vertical and is initially at its natural length with B at the same horizontal level as the axis. In this position a particle P is attached to the highest point of the pulley. This initial position is shown in Fig. 4.1. The radius of the pulley is \(a\), the mass of P is \(m\) and the stiffness of the string AB is \(\frac { m g } { 10 a }\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Fig. 4.2 shows the system with the pulley rotated through an angle \(\theta\) and the string stretched. Write down the extension of the string and hence find the potential energy, \(V\), of the system in this position. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \left( \frac { 1 } { 10 } \theta - \sin \theta \right)\).
2. Hence deduce that the system has a position of unstable equilibrium at \(\theta = 0\).
3. Explain how your expression for \(V\) relies on smooth contact between the string and the pulley. Fig. 4.3 shows the graph of the function \(\mathrm { f } ( \theta ) = \frac { 1 } { 10 } \theta - \sin \theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_538_1342_1706_404} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
4. Use the graph to give rough estimates of three further values of \(\theta\) (other than \(\theta = 0\) ) which give positions of equilibrium. In each case, state with reasons whether the equilibrium is stable or unstable.
5. Show on a sketch the physical situation corresponding to the least value of \(\theta\) you identified in part (iv). On your sketch, mark clearly the positions of P and B .
6. The equation \(\mathrm { f } ( \theta ) = 0\) has another root at \(\theta \approx - 2.9\). Explain, with justification, whether this necessarily gives a position of equilibrium.