OCR MEI M4 (Mechanics 4) 2015 June

1 A rocket is launched vertically upwards from rest. The initial mass of the rocket, including fuel and payload, is \(m _ { 0 }\) and the propulsion system ejects mass at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. The only other force acting on the rocket is its weight. The acceleration due to gravity is constant throughout the motion. At time \(t\) after launch the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket \(v = u \ln \left( \frac { m _ { 0 } } { m _ { 0 } - k t } \right) - g t\). The rocket initially has 2400 kg of fuel which is ejected at a constant rate of \(100 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) with constant speed \(3000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket.
2. Given that the rocket must reach a speed of \(7910 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before releasing its payload, find the maximum possible value of \(m _ { 0 }\).

2 Fig. 2 shows a system in a vertical plane. A uniform rod AB of length \(2 a\) and mass \(m\) is freely hinged at A . The angle that AB makes with the horizontal is \(\theta\), where \(- \frac { 2 } { 3 } \pi < \theta < \frac { 2 } { 3 } \pi\). Attached at B is a light spring BC of natural length \(a\) and stiffness \(\frac { m g } { a }\). The other end of the spring is attached to a small light smooth ring C which can slide freely along a vertical rail. The rail is at a distance of \(a\) from A and the spring is always horizontal. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system and hence show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \cos \theta ( 1 - 4 \sin \theta )\).
2. Hence find the positions of equilibrium of the system and investigate their stability.

3 A particle of mass 4 kg moves along the \(x\)-axis. At time \(t\) seconds the particle is \(x \mathrm {~m}\) from the origin O and has velocity \(v \mathrm {~ms} ^ { - 1 }\). A driving force of magnitude \(20 t \mathrm { t } ^ { - t } \mathrm {~N}\) is applied in the positive \(x\) direction. Initially \(v = 2\) and the particle is at O .

1. Find, in either order, the impulse of the force over the first 3 seconds and the velocity of the particle after 3 seconds. From time \(t = 3\) a resistive force of magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\) and the driving force are applied until the particle comes to rest.
2. Show that, after the resistive force is applied, the only time at which the resultant force on the particle is zero is when \(t = \ln 40\). Hence find the maximum velocity of the particle during the motion.
3. Given that the time \(T\) seconds at which the particle comes to rest is given by the equation \(T = \sqrt { 121 - 80 \mathrm { e } ^ { - T } ( 1 + T ) }\), without solving the equation deduce that \(T \approx 11\).
4. Use a numerical method to find \(T\) correct to 4 decimal places.

4 A solid cylinder of radius \(a \mathrm {~m}\) and length \(3 a \mathrm {~m}\) has density \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) given by \(\rho = k \left( 2 + \frac { x } { a } \right)\) where \(x \mathrm {~m}\) is the distance from one end and \(k\) is a positive constant. The mass of the cylinder is \(M \mathrm {~kg}\) where \(M = \frac { 21 } { 2 } \pi a ^ { 3 } k\). Let A and B denote the circular faces of the cylinder where \(x = 0\) and \(x = 3 a\), respectively.

1. Show by integration that the moment of inertia, \(I _ { \mathrm { A } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face A is given by \(I _ { \mathrm { A } } = \frac { 109 } { 28 } M a ^ { 2 }\).
2. Show that the centre of mass of the cylinder is \(\frac { 12 } { 7 } a \mathrm {~m}\) from A .
3. Using the parallel axes theorem, or otherwise, show that the moment of inertia, \(I _ { \mathrm { B } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face B is given by \(I _ { \mathrm { B } } = \frac { 73 } { 28 } M a ^ { 2 }\). You are now given that \(M = 4\) and \(a = 0.7\). The cylinder is at rest and can rotate freely about a horizontal axis which is a diameter of the face B as shown in Fig. 4. It is struck at the bottom of the curved surface by a small object of mass 0.2 kg which is travelling horizontally at speed \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical plane which is both perpendicular to the axis of rotation and contains the axis of symmetry of the cylinder. The object sticks to the cylinder at the point of impact. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-4_606_435_1087_817} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Find the initial angular speed of the combined object after the collision. \section*{END OF QUESTION PAPER}