Questions — OCR MEI M4 (44 questions)

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OCR MEI M4 2012 June Q2
2 A light elastic string AB has stiffness \(k\). The end A is attached to a fixed point and a particle of mass \(m\) is attached at the end B . With the string vertical, the particle is released from rest from a point at a distance \(a\) below its equilibrium position. At time \(t\), the displacement of the particle below the equilibrium position is \(x\) and the velocity of the particle is \(v\).
  1. Show that $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - k x$$
  2. Show that, while the particle is moving upwards and the string is taut, $$v = - \sqrt { \frac { k } { m } \left( a ^ { 2 } - x ^ { 2 } \right) }$$
  3. Hence use integration to find an expression for \(x\) at time \(t\) while the particle is moving upwards and the string is taut.
OCR MEI M4 2012 June Q3
3 A uniform rigid rod AB of length \(2 a\) and mass \(m\) is smoothly hinged to a fixed point at A so that it can rotate freely in a vertical plane. A light elastic string of modulus \(\lambda\) and natural length \(a\) connects the midpoint of AB to a fixed point C which is vertically above A with \(\mathrm { AC } = a\). The rod makes an angle \(2 \theta\) with the upward vertical, where \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \pi\). This is shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c4d3b482-5d09-4128-891d-4499fa49670c-3_339_563_534_737} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the potential energy, \(V\), of the system relative to A in terms of \(m , \lambda , a\) and \(\theta\). Show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 a \cos \theta ( 2 \lambda \sin \theta - 2 m g \sin \theta - \lambda ) .$$ Assume now that the system is set up so that the result (*) continues to hold when \(\pi < 2 \theta \leqslant \frac { 5 } { 3 } \pi\).
  2. In the case \(\lambda < 2 m g\), show that there is a stable position of equilibrium at \(\theta = \frac { 1 } { 2 } \pi\). Show that there are no other positions of equilibrium in this case.
  3. In the case \(\lambda > 2 m g\), find the positions of equilibrium for \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \frac { 5 } { 3 } \pi\) and determine for each whether the equilibrium is stable or unstable, justifying your conclusions.
OCR MEI M4 2012 June Q4
4
  1. Show by integration that the moment of inertia of a uniform circular lamina of radius \(a\) and mass \(m\) about an axis perpendicular to the plane of the lamina and through its centre is \(\frac { 1 } { 2 } m a ^ { 2 }\). A closed hollow cylinder has its curved surface and both ends made from the same uniform material. It has mass \(M\), radius \(a\) and height \(h\).
  2. Show that the moment of inertia of the cylinder about its axis of symmetry is \(\frac { 1 } { 2 } M a ^ { 2 } \left( \frac { a + 2 h } { a + h } \right)\). For the rest of this question take the cylinder to have mass 8 kg , radius 0.5 m and height 0.3 m .
    The cylinder is at rest and can rotate freely about its axis of symmetry. It is given a tangential impulse of magnitude 55 Ns at a point on its curved surface. The impulse is perpendicular to the axis.
  3. Find the angular speed of the cylinder after the impulse. A resistive couple is now applied to the cylinder for 5 seconds. The magnitude of the couple is \(2 \dot { \theta } ^ { 2 } \mathrm { Nm }\), where \(\dot { \theta }\) is the angular speed of the cylinder in rad s \({ } ^ { - 1 }\).
  4. Formulate a differential equation for \(\dot { \theta }\) and hence find the angular speed of the cylinder at the end of the 5 seconds. The cylinder is now brought to rest by a constant couple of magnitude 0.03 Nm .
  5. Calculate the time it takes from when this couple is applied for the cylinder to come to rest.
OCR MEI M4 2013 June Q1
1 An empty railway truck of mass \(m _ { 0 }\) is moving along a straight horizontal track at speed \(v _ { 0 }\). The point P is at the front of the truck. The horizontal forces on the truck are negligible. As P passes a fixed point O , sand starts to fall vertically into the truck at a constant mass rate \(k\). At time \(t\) after P passes O the speed of the truck is \(v\) and \(\mathrm { OP } = x\).
  1. Find an expression for \(v\) in terms of \(m _ { 0 } , v _ { 0 } , k\) and \(t\), and show that \(x = \frac { m _ { 0 } v _ { 0 } } { k } \ln \left( 1 + \frac { k t } { m _ { 0 } } \right)\).
  2. Find the speed of the truck and the distance OP when the mass of sand in the truck is \(2 m _ { 0 }\).
OCR MEI M4 2013 June Q2
2 A uniform rod AB of length 0.5 m and mass 0.5 kg is freely hinged at A so that it can rotate in a vertical plane. Attached at B are two identical light elastic strings BC and BD each of natural length 0.5 m and stiffness \(2 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). The ends C and D are fixed at the same horizontal level as A and with \(\mathrm { AC } = \mathrm { CD } = 0.5 \mathrm {~m}\). The system is shown in Fig. 2.1 with the angle \(\mathrm { BAC } = \theta\). You may assume that \(\frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 5 } { 3 } \pi\) so that both strings are taut. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-2_328_732_1032_667} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Show that the length of BC in metres is \(\sin \frac { 1 } { 2 } \theta\).
  2. Find the potential energy, \(V \mathrm {~J}\), of the system relative to AD in terms of \(\theta\). Hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta .$$
  3. Fig. 2.2 shows a graph of the function \(\mathrm { f } ( \theta ) = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-2_453_1264_2021_397} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} Use the graph both to estimate, correct to 1 decimal place, the values of \(\theta\) for which the system is in equilibrium and also to determine their stability.
OCR MEI M4 2013 June Q3
3 A model car of mass 2 kg moves from rest along a horizontal straight path. After time \(t \mathrm {~s}\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power, \(P \mathrm {~W}\), developed by the engine is initially modelled by \(P = 2 v ^ { 3 } + 4 v\). The car is subject to a resistance force of magnitude \(6 v \mathrm {~N}\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = ( 1 - v ) ( 2 - v )\) and hence show that \(t = \ln \frac { 2 - v } { 2 ( 1 - v ) }\).
  2. Hence express \(v\) in terms of \(t\). Once the power reaches 4.224 W it remains at this constant value with the resistance force still acting.
  3. Verify that the power of 4.224 W is reached when \(v = 0.8\) and calculate the value of \(t\) at this instant.
  4. Find \(v\) in terms of \(t\) for the motion at constant power. Deduce the limiting value of \(v\) as \(t \rightarrow \infty\).
OCR MEI M4 2013 June Q4
4 A uniform lamina of mass \(m\) is in the shape of a sector of a circle of radius \(a\) and angle \(\frac { 1 } { 3 } \pi\). It can rotate freely in a vertical plane about a horizontal axis perpendicular to the lamina through its vertex O .
  1. Show by integration that the moment of inertia of the lamina about the axis is \(\frac { 1 } { 2 } m a ^ { 2 }\).
  2. State the distance of the centre of mass of the lamina from the axis. The lamina is released from rest when one of the straight edges is horizontal as shown in Fig. 4.1. After time \(t\), the line of symmetry of the lamina makes an angle \(\theta\) with the downward vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_257_441_1475_322} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-3_380_732_1635_1014} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
  3. Show that \(\dot { \theta } ^ { 2 } = \frac { 4 g } { \pi a } ( 2 \cos \theta + 1 )\).
  4. Find the greatest speed attained by any point on the lamina.
  5. Find an expression for \(\ddot { \theta }\) in terms of \(\theta , a\) and \(g\). The lamina strikes a fixed peg at A where \(\mathrm { AO } = \frac { 3 } { 4 } a\) and is horizontal, as shown in Fig. 4.2. The collision reverses the direction of motion of the lamina and halves its angular speed.
  6. Find the magnitude of the impulse that the peg gives to the lamina.
  7. Determine the maximum value of \(\theta\) in the subsequent motion.
OCR MEI M4 2014 June Q1
1 A sports car of mass 1.2 tonnes is being tested on a horizontal, straight section of road. After \(t \mathrm {~s}\), the car has travelled \(x \mathrm {~m}\) from the starting line and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine produces a driving force of 4000 N and the total resistance to the motion of the car is given by \(\frac { 40 } { 49 } v ^ { 2 } \mathrm {~N}\). The car crosses the starting line with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down an equation of motion for the car and solve it to show that \(v ^ { 2 } = 4900 - 4800 \mathrm { e } ^ { - \frac { 1 } { 735 } x }\).
  2. Hence find the work done against the resistance to motion over the first 100 m beyond the starting line.
OCR MEI M4 2014 June Q2
2 On a building site, a pulley system is used for moving small amounts of material up to roof level. A light pulley, which can rotate freely, is attached with its axis horizontal to the top of some scaffolding. A light inextensible rope hangs over the pulley with a counterweight of mass \(m _ { 1 } \mathrm {~kg}\) attached to one end. Attached to the other end of the rope is a bag of negligible mass into which \(m _ { 2 } \mathrm {~kg}\) of roof tiles are placed, where \(m _ { 2 } < m _ { 1 }\). This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-2_554_711_1098_678} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Initially the system is held at rest with the rope taut, the counterweight at the top of the scaffolding and the bag of tiles on the ground. When the counterweight is released, the bag ascends towards the top of the scaffolding. At time \(t \mathrm {~s}\) the velocity of the counterweight is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. The counterweight is made from a bag of negligible mass filled with sand. At the moment the counterweight is released, this bag is accidentally ripped and after this time the sand drops out at a constant rate of \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\).
  1. Find the equation of motion for the counterweight while it still contains sand, and hence show that $$v = g t + \frac { 2 g m _ { 2 } } { \lambda } \ln \left( 1 - \frac { \lambda t } { m _ { 1 } + m _ { 2 } } \right) .$$
  2. Given that the sand would run out after 10 seconds and that \(m _ { 2 } = \frac { 4 } { 5 } m _ { 1 }\), find the maximum velocity attained by the counterweight towards the ground. You may assume that the scaffolding is sufficiently high that the counterweight does not hit the ground before this velocity is reached.
OCR MEI M4 2014 June Q3
3 A uniform rigid rod AB of mass \(m\) and length \(2 a\) is freely hinged to a horizontal floor at A . The end B is attached to a light elastic string of modulus \(\lambda\) and natural length \(5 a\). The other end of the string is attached to a small, light, smooth ring C which can slide along a horizontal rail. The rail is a distance \(7 a\) above the floor and C is always vertically above B . The angle that AB makes with the floor is \(\theta\). The system is shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-3_664_773_584_648} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the potential energy, \(V\), of the system and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( m g - \frac { 4 \lambda } { 5 } ( 1 - \sin \theta ) \right) .$$
  2. Show that there is a position of equilibrium when \(\theta = \frac { 1 } { 2 } \pi\) and determine whether or not it is stable. There are two further positions of equilibrium when \(0 < \theta < \pi\).
  3. Find the magnitude of the tension in the string and the vertical force of the hinge on the rod in these positions.
  4. Show that \(\lambda > \frac { 5 m g } { 4 }\).
  5. Show that these positions of equilibrium are stable.
OCR MEI M4 2014 June Q4
4
  1. A pulley consists of a central cylinder of wood and an outer ring of steel. The density of the wood is \(700 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the density of the steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). The pulley has a radius of 20 cm and is 10 cm thick (see Fig. 4.1). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-4_359_661_404_742} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure} Find the radius that the central cylinder must have in order that the moment of inertia of the pulley about the axis of symmetry shown in Fig. 4.1 is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
  2. Two blocks P and Q of masses 10 kg and 20 kg are connected by a light inextensible string. The string passes over a heavy rough pulley of radius 25 cm . The pulley can rotate freely and the string does not slip. Block P is held at rest in smooth contact with a plane inclined at \(30 ^ { \circ }\) to the horizontal, and block Q is at rest below the pulley (see Fig. 4.2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-5_341_917_438_541} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure} At \(t \mathrm {~s}\) after the system is released from rest, the pulley has angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and block P has constant acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up the slope.
    1. Show that the net loss of energy of the two blocks in the first \(t\) seconds of motion is \(87 t ^ { 2 } \mathrm {~J}\) and use the principle of conservation of energy to show that the moment of inertia of the pulley about its axis of rotation is \(\frac { 87 } { 32 } \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When \(t = 3\) a resistive couple is applied to the pulley. This resistive couple has magnitude \(( 2 \omega + k ) \mathrm { Nm }\), where \(k\) is a constant. The couple on the pulley due to tensions in the sections of string is \(\left( \frac { 147 } { 4 } - \frac { 15 } { 8 } \frac { \mathrm {~d} \omega } { \mathrm {~d} t } \right) \mathrm { Nm }\) in the direction of positive \(\omega\).
    2. Write down a first order differential equation for \(\omega\) when \(t \geqslant 3\) and show by integration that $$\omega = \frac { 1 } { 8 } \left( ( 45 + 4 k ) \mathrm { e } ^ { \frac { 64 } { 147 } ( 3 - t ) } + 147 - 4 k \right) .$$
    3. By considering the equation given in part (ii), find the value or set of values of \(k\) for which the pulley
      (A) continues to rotate with constant angular velocity,
      (B) rotates with decreasing angular velocity without coming to rest,
      (C) rotates with decreasing angular velocity and comes to rest if there is sufficient distance between P and the pulley. \section*{END OF QUESTION PAPER}
OCR MEI M4 2015 June Q1
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 \(m _ { 0 }\).
OCR MEI M4 2015 June Q2
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]
\includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-2_737_703_1356_680} \captionsetup{labelformat=empty} \caption{Fig. 2}
\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.
OCR MEI M4 2015 June Q3
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.
OCR MEI M4 2015 June Q4
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}
OCR MEI M4 2016 June Q1
1 A car of mass \(m\) moves horizontally in a straight line. At time \(t\) the car is a distance \(x\) from a point O and is moving away from O with speed \(v\). There is a force of magnitude \(k v ^ { 2 }\), where \(k\) is a constant, resisting the motion of the car. The car's engine has a constant power \(P\). The terminal speed of the car is \(U\).
  1. Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
  2. Show that the distance moved while the car accelerates from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\) is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant \(A\). Once the car attains a speed of \(\frac { 1 } { 2 } U\), no further power is supplied by the car's engine.
  3. Find, in terms of \(m , P\) and \(U\), the time taken for the speed of the car to reduce from \(\frac { 1 } { 2 } U\) to \(\frac { 1 } { 4 } U\).
OCR MEI M4 2016 June Q2
2 A thin rigid rod PQ has length \(2 a\). Its mass per unit length, \(\rho\), is given by \(\rho = k \left( 1 + \frac { x } { 2 a } \right)\) where \(x\) is the distance from P and \(k\) is a positive constant. The mass of the rod is \(M\) and the moment of inertia of the rod about an axis through P perpendicular to PQ is \(I\).
  1. Show that \(I = \frac { 14 } { 9 } M a ^ { 2 }\). The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where \(x = \frac { 3 } { 2 } a\). The magnitude of the impulse of this blow is \(J\).
  2. Find, in terms of \(a , J\) and \(M\), the initial angular speed of the rod.
  3. Find, in terms of \(a , g\) and \(M\), the set of values of \(J\) for which the rod makes complete revolutions.
OCR MEI M4 2016 June Q3
3 Fig. 3 shows a uniform rigid rod AB of length \(2 a\) and mass \(2 m\). The rod is freely hinged at A so that it can rotate in a vertical plane. One end of a light inextensible string of length \(l\) is attached to B . The string passes over a small smooth fixed pulley at C , where C is vertically above A and \(\mathrm { AC } = 6 a\). A particle of mass \(\lambda m\), where \(\lambda\) is a positive constant, is attached to the other end of the string and hangs freely, vertically below C . The rod makes an angle \(\theta\) with the upward vertical, where \(0 \leqslant \theta \leqslant \pi\). You may assume that the particle does not interfere with the rod AB or the section of the string BC . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fdb2cff-0f74-4c88-b25a-759bfab1675a-3_878_615_667_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the potential energy, \(V\), of the system relative to a situation in which the rod AB is horizontal, and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \sin \theta \left( \frac { 3 \lambda } { \sqrt { 10 - 6 \cos \theta } } - 1 \right) .$$
  2. Show that \(\theta = 0\) and \(\theta = \pi\) are the only values of \(\theta\) for which the system is in equilibrium whatever the value of \(\lambda\).
  3. Show that, if there is a third value of \(\theta\) for which the system is in equilibrium, then \(\frac { 2 } { 3 } < \lambda < \frac { 4 } { 3 }\).
  4. Given that there are three positions of equilibrium, establish whether each of these positions is stable or unstable. It is given that, for small values of \(\theta\), $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } \approx 2 m g a \left[ \left( \frac { 3 } { 2 } \lambda - 1 \right) \theta - \left( \frac { 13 } { 16 } \lambda - \frac { 1 } { 6 } \right) \theta ^ { 3 } \right] .$$
  5. Investigate the stability of the equilibrium position given by \(\theta = 0\) in the case when \(\lambda = \frac { 2 } { 3 }\).
OCR MEI M4 2016 June Q4
4 A raindrop falls from rest through a stationary cloud. The raindrop has mass \(m\) and speed \(v\) when it has fallen a distance \(x\). You may assume that resistances to motion are negligible.
  1. Derive the equation of motion $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } + v ^ { 2 } \frac { \mathrm {~d} m } { \mathrm {~d} x } = m g .$$ Initially the mass of the raindrop is \(m _ { 0 }\). Two different models for the mass of the raindrop are suggested.
    In the first model \(m = m _ { 0 } \mathrm { e } ^ { k _ { 1 } x }\), where \(k _ { 1 }\) is a positive constant.
  2. Show that $$v ^ { 2 } = \frac { g } { k _ { 1 } } \left( 1 - \mathrm { e } ^ { - 2 k _ { 1 } x } \right) ,$$ and hence state, in terms of \(g\) and \(k _ { 1 }\), the terminal velocity of the raindrop according to this first model. In the second model \(m = m _ { 0 } \left( 1 + k _ { 2 } x \right)\), where \(k _ { 2 }\) is a positive constant.
  3. By considering the expression obtained from differentiating \(v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 }\) with respect to \(x\), show that, for the second model, the equation of motion in part (i) may be written as $$\frac { d } { d x } \left[ v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 } \right] = 2 g \left( 1 + k _ { 2 } x \right) ^ { 2 } .$$ Hence find an expression for \(v ^ { 2 }\) in terms of \(g , k _ { 2 }\) and \(x\).
  4. Suppose that the models give the same value for the speed of the raindrop at the instant when it has doubled its initial mass. Find the exact value of \(\frac { k _ { 1 } } { k _ { 2 } }\), giving your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\)
    are integers. are integers. \section*{END OF QUESTION PAPER}