OCR M4 (Mechanics 4) 2012 June

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
2. Find the time taken for the lamina to come to rest.

2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis and the curve \(y = x \left( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } \right)\) for \(0 \leqslant x \leqslant a\), where \(a\) is a constant, about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

3
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-2_460_388_1160_826} A ship \(S\) is travelling with constant velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(120 ^ { \circ }\). A patrol boat \(B\) observes the ship when \(S\) is due north of \(B\). The patrol boat \(B\) then moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line (see diagram).

1. Given that \(V = 18\), find the bearing of the course of \(B\) such that \(B\) intercepts \(S\).
2. Given instead that \(V = 9\), find the bearing of the course of \(B\) such that \(B\) passes as close as possible to \(S\).
3. Find the smallest value of \(V\) for which it is possible for \(B\) to intercept \(S\).

4 A uniform lamina of mass 18 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 9\) and the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln 9\). The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.

5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.

1. Find the angular acceleration of the rod immediately after it is released.
2. Find the angle that the rod makes with the horizontal when its angular acceleration is zero.
3. Find the maximum angular speed of the rod.
4. The rod first comes to instantaneous rest after rotating through an angle \(\theta\) radians from its initial position. Find an equation for \(\theta\), and verify that \(2.0 < \theta < 2.1\).

6
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs \(P\) and \(Q\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(A\) is the mid-point of \(P Q\). A light rod \(A B\) of length \(4 a\) is freely pivoted at \(A\) and can rotate in the vertical plane containing \(P Q\), with \(B\) below the level of \(P Q\). A particle of mass \(m\) is attached to the rod at \(B\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\lambda\), passes round the pegs \(P\) and \(Q\) and its two ends are attached to the rod at the point \(X\), where \(A X = a\). The angle between the rod and the downward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram). You are given that the elastic energy stored in the string is \(\lambda a ( 1 + \cos \theta )\).

1. Show that \(\theta = 0\) is a position of equilibrium, and show that the equilibrium is stable if \(\lambda < 4 m g\).
2. Given that \(\lambda = 3 m g\), show that \(\ddot { \theta } = - k \frac { g } { a } \sin \theta\), stating the value of the constant \(k\). Hence find the approximate period of small oscillations of the system about the equilibrium position \(\theta = 0\).

7
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre \(C\) has mass \(m\) and radius \(a\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(A C = \frac { 1 } { 2 } a\). The disc is slightly disturbed from rest in the position with \(C\) vertically above \(A\). When \(A C\) makes an angle \(\theta\) with the upward vertical the force exerted by the axis on the disc has components \(R\) parallel to \(A C\) and \(S\) perpendicular to \(A C\) (see diagram).

1. Show that the angular speed of the disc is \(\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }\).
2. Find the angular acceleration of the disc, in terms of \(a , g\) and \(\theta\).
3. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
4. Find the magnitude of the force exerted by the axis on the disc at an instant when \(R = 0\).