OCR M4 (Mechanics 4) 2004 January

1 A wheel is rotating about a fixed axis, and is slowing down with constant angular deceleration \(0.3 \mathrm { rad } \mathrm { s } ^ { - 2 }\).

1. Find the angle the wheel turns through as its angular speed changes from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the time taken for the wheel to make its final complete revolution before coming to rest.

2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).

3 From a speedboat, a ship is sighted on a bearing of \(045 ^ { \circ }\). The ship has constant velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(120 ^ { \circ }\). The speedboat travels in a straight line with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and intercepts the ship.

1. Find the bearing of the course of the speedboat.
2. Find the magnitude of the velocity of the ship relative to the speedboat.

4 The region between the curve \(y = \frac { x ^ { 2 } } { a }\) and the \(x\)-axis for \(0 \leqslant x \leqslant a\) is occupied by a uniform lamina with mass \(m\). Show that the moment of inertia of this lamina about the \(x\)-axis is \(\frac { 1 } { 7 } m a ^ { 2 }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-2_618_627_1594_743} A uniform circular disc has mass 4 kg , radius 0.6 m and centre \(C\). The disc can rotate in a vertical plane about a fixed horizontal axis which is perpendicular to the disc and which passes through the point \(A\) on the disc, where \(A C = 0.4 \mathrm {~m}\). A frictional couple of constant moment 4.8 Nm opposes the motion. The disc is released from rest with \(A C\) horizontal (see diagram).

1. Find the moment of inertia of the disc about the axis through \(A\).
2. Find the angular acceleration of the disc immediately after it is released.
3. Find the angular speed of the disc when \(C\) is first vertically below \(A\).

6 A rigid body consists of a uniform rod \(A B\), of mass 15 kg and length 2.8 m , with a particle of mass 5 kg attached at \(B\). The body rotates without resistance in a vertical plane about a fixed horizontal axis through \(A\).

1. Find the distance of the centre of mass of the body from \(A\).
2. Find the moment of inertia of the body about the axis.
   \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-3_475_682_680_719} At one instant, \(A B\) makes an acute angle \(\theta\) with the downward vertical, the angular speed of the body is \(1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular acceleration of the body is \(3.5 \mathrm { rad } \mathrm { s } ^ { - 2 }\) (see diagram).
3. Show that \(\sin \theta = 0.8\).
4. Find the components, parallel and perpendicular to \(B A\), of the force acting on the body at \(A\).
   [0pt] [Question 7 is printed overleaf.]
   \includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-4_949_1112_281_550} A small bead \(B\), of mass \(m\), slides on a smooth circular hoop of radius \(a\) and centre \(O\) which is fixed in a vertical plane. A light elastic string has natural length \(2 a\) and modulus of elasticity \(m g\); one end is attached to \(B\), and the other end is attached to a light ring \(R\) which slides along a smooth horizontal wire. The wire is in the same vertical plane as the hoop, and at a distance \(2 a\) above \(O\). The elastic string \(B R\) is always vertical, and \(O B\) makes an angle \(\theta\) with the downward vertical (see diagram).
5. Show that \(\theta = 0\) is a position of stable equilibrium.
6. Find the approximate period of small oscillations about the equilibrium position \(\theta = 0\).