OCR M4 (Mechanics 4) 2010 June

1 A wheel is rotating and is slowing down with constant angular deceleration. The initial angular speed is \(80 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and after 15 s the wheel has turned through 1020 radians.

1. Find the angular deceleration of the wheel.
2. Find the angle through which the wheel turns in the last 5 s before it comes to rest.
3. Find the total number of revolutions made by the wheel from the start until it comes to rest.

2 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 3\), and the curve \(y = \mathrm { e } ^ { - x }\) for \(0 \leqslant x \leqslant \ln 3\), is occupied by a uniform lamina. Find, in an exact form, the coordinates of the centre of mass of this lamina.

3 A circular disc is rotating in a horizontal plane with angular speed \(16 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis passing through its centre \(O\). The moment of inertia of the disc about the axis is \(0.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A particle, initially at rest just above the surface of the disc, drops onto the disc and sticks to it at a point 0.4 m from \(O\). Afterwards, the angular speed of the disc with the particle attached is \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle.
2. Find the loss of kinetic energy.

4
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-2_688_777_1382_683} From a boat \(B\), a cruiser \(C\) is observed 3500 m away on a bearing of \(040 ^ { \circ }\). The cruiser \(C\) is travelling with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line course with bearing \(110 ^ { \circ }\) (see diagram). The boat \(B\) travels with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight line course which takes it as close as possible to the cruiser \(C\).

1. Show that the bearing of the course of \(B\) is \(073 ^ { \circ }\), correct to the nearest degree.
2. Find the magnitude and the bearing of the velocity of \(C\) relative to \(B\).
3. Find the shortest distance between \(B\) and \(C\) in the subsequent motion.

5 A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The point \(C\) on the rod is such that \(A C = a\). The rod can rotate freely in a vertical plane about a fixed horizontal axis passing through \(C\) and perpendicular to the rod.

1. Show by integration that the moment of inertia of the rod about this axis is \(7 m a ^ { 2 }\). The rod starts at rest with \(B\) vertically below \(C\). A couple of constant moment \(\frac { 6 m g a } { \pi }\) is then applied to the rod.
2. Find, in terms of \(a\) and \(g\), the angular speed of the rod when it has turned through one and a half revolutions.
   \includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-3_721_621_872_762} A light pulley of radius \(a\) is free to rotate in a vertical plane about a fixed horizontal axis passing through its centre \(O\). Two particles, \(P\) of mass \(5 m\) and \(Q\) of mass \(3 m\), are connected by a light inextensible string. The particle \(P\) is attached to the circumference of the pulley, the string passes over the top of the pulley, and \(Q\) hangs below the pulley on the opposite side to \(P\). The section of string not in contact with the pulley is vertical. The fixed line \(O X\) makes an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(O P\) makes an angle \(\theta\) with \(O X\) (see diagram). You are given that the total potential energy of the system (using a suitable reference level) is \(V\), where $$V = m g a ( 3 \sin \theta - 4 \cos \theta - 3 \theta ) .$$

1. Show that \(\theta = 0\) is a position of stable equilibrium.
2. Show that the kinetic energy of the system is \(4 m a ^ { 2 } \dot { \theta } ^ { 2 }\).
3. By differentiating the energy equation, then making suitable approximations for \(\sin \theta\) and \(\cos \theta\), find the approximate period of small oscillations about the equilibrium position \(\theta = 0\). \section*{[Question 7 is printed overleaf.]}

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\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina \(A B C D\) with \(A B = 6 a , A D = 8 a\) and centre \(G\). The mass of the lamina is \(m\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\) and perpendicular to the lamina.

1. Find the moment of inertia of the lamina about this axis. The lamina is released from rest with \(A D\) horizontal and \(B C\) below \(A D\).
2. For an instant during the subsequent motion when \(A D\) is vertical, show that the angular speed of the lamina is \(\sqrt { \frac { 3 g } { 50 a } }\) and find its angular acceleration. At an instant when \(A D\) is vertical, the force acting on the lamina at \(A\) has magnitude \(F\).
3. By finding components parallel and perpendicular to \(G A\), or otherwise, show that \(F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }\).
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