OCR M4 (Mechanics 4) 2006 June

1 A straight rod \(A B\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k ( a + 2 x )\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod.

2 A flywheel takes the form of a uniform disc of mass 8 kg and radius 0.15 m . It rotates freely about an axis passing through its centre and perpendicular to the disc. A couple of constant moment is applied to the flywheel. The flywheel turns through an angle of 75 radians while its angular speed increases from \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the moment of the couple about the axis. When the flywheel is rotating with angular speed \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\), it locks together with a second flywheel which is mounted on the same axis and is at rest. Immediately afterwards, both flywheels rotate together with the same angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the moment of inertia of the second flywheel about the axis.

3 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), is occupied by a uniform lamina of mass 24 kg . The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-2_623_601_1409_706} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2 m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5 m g\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(A B\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(A B\) and the ring \(R\) are in a vertical plane, and \(R C\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac { 1 } { 4 } \pi\), so that the acute angle between \(A B\) and the horizontal is \(2 \theta\) (see diagram).

1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium.
2. Determine whether this position of equilibrium is stable or unstable.

5 A uniform rectangular lamina \(A B C D\) has mass 20 kg and sides of lengths \(A B = 0.6 \mathrm {~m}\) and \(B C = 1.8 \mathrm {~m}\). It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(A B\).

1. Show that the moment of inertia of the lamina about the axis is \(22.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_442_541_477_800} The lamina is released from rest with \(B C\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 Nm about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).
2. Show that the angular acceleration is zero when \(\cos \theta = 0.25\).
3. Hence find the maximum angular speed of the lamina.
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-3_633_838_1356_650} A ship \(P\) is moving with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A second ship \(Q\) is moving with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of \(050 ^ { \circ }\) (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m .
4. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), to the nearest degree, and find the bearing of the other possible direction of this relative velocity. Given that the velocity of \(Q\) relative to \(P\) has bearing \(036 ^ { \circ }\), find
5. the bearing of the direction in which \(Q\) is moving,
6. the magnitude of the velocity of \(Q\) relative to \(P\),
7. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together,
8. the bearing of \(P\) from \(Q\) when the two ships are closest together.
   \includegraphics[max width=\textwidth, alt={}, center]{d5c6deb0-ef1a-4878-889d-dc9f926aaf88-4_560_1180_265_467} A uniform rod \(A B\) has mass \(m\) and length \(6 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(A C = a\). The angle between \(A B\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(A B\) and \(S\) perpendicular to \(A B\) (see diagram). The rod is released from rest in the position where \(\theta = \frac { 1 } { 3 } \pi\). Air resistance may be neglected.
9. Find the angular acceleration of the rod in terms of \(a , g\) and \(\theta\).
10. Show that the angular speed of the rod is \(\sqrt { \frac { 2 g ( 1 - 2 \cos \theta ) } { 7 a } }\).
11. Find \(R\) and \(S\) in terms of \(m , g\) and \(\theta\).
12. When \(\cos \theta = \frac { 1 } { 3 }\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude.