OCR M4 (Mechanics 4) 2005 June

1 A wheel is rotating freely with angular speed \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed axis through its centre. The moment of inertia of the wheel about the axis is \(0.65 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A couple of constant moment is applied to the wheel, and in the next 5 seconds the wheel rotates through 180 radians.

1. Find the angular acceleration of the wheel.
2. Find the moment of the couple about the axis.

2 The region enclosed by the curve \(y = \sqrt { } x\) for \(0 \leqslant x \leqslant 9\), the \(x\)-axis and the line \(x = 9\) is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

3
\includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-2_653_406_727_857} A lamina has mass 1.5 kg . Two perpendicular lines \(A B\) and \(C D\) in the lamina intersect at the point \(X\). The centre of mass, \(G\), of the lamina lies on \(A B\), and \(X G = 0.2 \mathrm {~m}\) (see diagram). The moment of inertia of the lamina about \(A B\) is \(0.02 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the moment of inertia of the lamina about \(C D\) is \(0.12 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis perpendicular to the lamina and passing through \(X\).

1. The lamina makes small oscillations as a compound pendulum. Find the approximate period of these oscillations.
2. The lamina starts at rest with \(G\) vertically below \(X\). A couple of constant moment 3.2 Nm about the axis is now applied to the lamina. Find the angular speed of the lamina when \(X G\) is first horizontal.

4 A boat \(A\) has constant velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A boat \(B\), which is initially 250 m due south of \(A\), moves with constant speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction which takes it as close as possible to \(A\).

1. Find the bearing of the direction in which \(B\) moves.
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.

5 In this question, \(a\) and \(k\) are positive constants.
The region enclosed by the curve \(y = a \mathrm { e } ^ { - \frac { x } { a } }\) for \(0 \leqslant x \leqslant k a\), the \(x\)-axis, the \(y\)-axis and the line \(x = k a\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of mass \(m\). Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 1 } { 4 } m a ^ { 2 } \left( 1 + \mathrm { e } ^ { - 2 k } \right)\).

6 A uniform circular disc, of mass \(m\) and radius \(a\), has centre \(C\). The disc can rotate freely in a vertical plane about a fixed horizontal axis through the point \(A\) on the disc, where \(C A = \frac { 1 } { 2 } a\). The disc is released from rest in the position with \(C A\) horizontal. When the disc has rotated through an angle \(\theta\),

1. show that the angular acceleration of the disc is \(\frac { 2 g \cos \theta } { 3 a }\),
2. find the angular speed of the disc,
3. find the components, parallel and perpendicular to \(C A\), of the force acting on the disc at the axis.

7
\includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-3_585_801_991_647} A light rod \(A B\) of length \(2 a\) can rotate freely in a vertical plane about a fixed horizontal axis through \(A\). A particle of mass \(m\) is attached to the rod at \(B\). A fixed smooth ring \(R\) lies in the same vertical plane as the rod, where \(A R = a\) and \(A R\) makes an angle \(\frac { 1 } { 4 } \pi\) above the horizontal. A light elastic string, of natural length \(a\) and modulus of elasticity \(m g \sqrt { } 2\), passes through the ring \(R\); one end is fixed to \(A\) and the other end is fixed to \(B\). The rod makes an angle \(\theta\) below the horizontal, where \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\) (see diagram).

1. Use the cosine rule to show that \(R B ^ { 2 } = a ^ { 2 } ( 5 - ( 2 \sqrt { } 2 ) \cos \theta + ( 2 \sqrt { } 2 ) \sin \theta )\).
2. Show that \(\theta = 0\) is a position of stable equilibrium.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - k \sin \theta\), expressing the constant \(k\) in terms of \(a\) and \(g\), and hence write down the approximate period of small oscillations about the equilibrium position \(\theta = 0\).