OCR MEI M4 (Mechanics 4) 2011 June

1 A raindrop of mass \(m\) falls vertically from rest under gravity. Initially the mass of the raindrop is \(m _ { 0 }\). As it falls it loses mass by evaporation at a rate \(\lambda m\), where \(\lambda\) is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is \(v\) at time \(t\). The forces on the raindrop are its weight and a resistance force of magnitude \(k m v\), where \(k\) is a constant.

1. Find \(m\) in terms of \(m _ { 0 } , \lambda\) and \(t\).
2. Write down the equation of motion of the raindrop. Solve this differential equation and hence show that \(v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)\).
3. Find the velocity of the raindrop when it has lost half of its initial mass.

2 A small ring of mass \(m\) can slide freely along a fixed smooth horizontal rod. A light elastic string of natural length \(a\) and stiffness \(k\) has one end attached to a point A on the rod and the other end attached to the ring. An identical elastic string has one end attached to the ring and the other end attached to a point B which is a distance \(a\) vertically above the rod and a horizontal distance \(2 a\) from the point A . The displacement of the ring from the vertical line through B is \(x\), as shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find an expression for \(V\), the potential energy of the system when \(0 < x < a\), and show that $$\frac { \mathrm { d } V } { \mathrm {~d} x } = 2 k x - k a - \frac { k a x } { \sqrt { a ^ { 2 } + x ^ { 2 } } }$$
2. Show that \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) is always positive.
3. Show that there is a position of equilibrium with \(\frac { 1 } { 2 } a < x < a\). State, with a reason, whether it is stable or unstable.

3 A car of mass 800 kg moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds. While \(v \leqslant 20\), the power developed by the engine is \(8 v ^ { 4 } \mathrm {~W}\). The total resistance force on the car is of magnitude \(8 v ^ { 2 } \mathrm {~N}\). Initially \(v = 2\) and the car is at a point O . At time \(t\) s the displacement from O is \(x \mathrm {~m}\).

1. Find \(v\) in terms of \(x\) and show that when \(v = 20 , x = 100 \ln 1.9\).
2. Find the relationship between \(t\) and \(x\), and show that when \(v = 20 , t \approx 19.2\). The driving force is removed at the instant when \(v\) reaches 20 .
3. For the subsequent motion, find \(v\) in terms of \(t\). Calculate \(t\) when \(v = 2\).

4 In this question you may assume without proof the standard results in Examination Formulae and Tables (MF2) for

• the moment of inertia of a disc about an axis through its centre perpendicular to the disc,
• the position of the centre of mass of a solid uniform cone.

Fig. 4 shows a uniform cone of radius \(a\) and height \(2 a\), with its axis of symmetry on the \(x\)-axis and its vertex at the origin. A thin slice through the cone parallel to the base is at a distance \(x\) from the vertex. \begin{figure}[h] \end{figure} The slice is taken to be a thin uniform disc of mass \(m\).

1. Write down the moment of inertia of the disc about the \(x\)-axis. Hence show that the moment of inertia of the disc about the \(y\)-axis is \(\frac { 17 } { 16 } m x ^ { 2 }\).
2. Hence show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 51 } { 20 } M a ^ { 2 }\), where \(M\) is the mass of the cone. [You may assume without proof the formula for the volume of a cone.] The cone is now suspended so that it can rotate freely about a fixed, horizontal axis through its vertex. The axis of symmetry of the cone moves in a vertical plane perpendicular to the axis of rotation. The cone is released from rest when its axis of symmetry is at an acute angle \(\alpha\) to the downward vertical. At time \(t\), the angle the axis of symmetry makes with the downward vertical is \(\theta\).
3. Use an energy method to show that \(\dot { \theta } ^ { 2 } = \frac { 20 g } { 17 a } ( \cos \theta - \cos \alpha )\).
4. Hence, or otherwise, show that if \(\alpha\) is small the cone performs approximate simple harmonic motion and find the period. RECOGNISING ACHIEVEMENT