OCR M4 (Mechanics 4) 2009 June

1 A top is set spinning with initial angular speed \(83 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it slows down with constant angular deceleration. When it has turned through 1000 radians, its angular speed is \(67 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the angular deceleration of the top.
2. Find the time taken, from the start, for the top to turn through 400 radians.

2 The region \(R\) is bounded by the \(x\)-axis, the lines \(x = a\) and \(x = 2 a\), and the curve \(y = \frac { a ^ { 3 } } { x ^ { 2 } }\) for \(a \leqslant x \leqslant 2 a\), where \(a\) is a positive constant. A uniform solid of revolution is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.

3
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-2_664_623_904_760} A uniform circular disc has mass \(4 m\), radius \(2 a\) and centre \(O\). The points \(A\) and \(B\) are at opposite ends of a diameter of the disc, and the mid-point of \(O A\) is \(P\). A particle of mass \(m\) is attached to the disc at \(B\). The resulting compound pendulum is in a vertical plane and is free to rotate about a fixed horizontal axis passing through \(P\) and perpendicular to the disc (see diagram). The pendulum makes small oscillations.

1. Find the moment of inertia of the pendulum about the axis.
2. Find the approximate period of the small oscillations.

4 From a helicopter, a small plane is spotted 3750 m away on a bearing of \(075 ^ { \circ }\). The plane is at the same altitude as the helicopter, and is flying with constant speed \(62 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal straight line on a bearing of \(295 ^ { \circ }\). The helicopter flies with constant speed \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line, and intercepts the plane.

1. Find the bearings of the two possible directions in which the helicopter could fly.
2. Given that interception occurs in the shorter of the two possible times, find the time taken to make the interception.
   \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-3_668_298_260_922} A uniform lamina of mass 63 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, and the curve \(y = 8 - x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\). The unit of length is the metre. The vertices of the lamina are \(O ( 0,0 )\), \(A ( 2,0 )\) and \(B ( 0,8 )\) (see diagram).
3. Show that the moment of inertia of this lamina about \(O B\) is \(56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). It is given that the moment of inertia of the lamina about \(O A\) is \(1036.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the centre of mass of the lamina has coordinates \(\left( \frac { 4 } { 5 } , \frac { 24 } { 7 } \right)\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis passing through \(O\) and perpendicular to the lamina. Starting with the lamina at rest with \(B\) vertically above \(O\), a couple of constant anticlockwise moment 800 Nm is applied to the lamina.
4. Show that the lamina begins to rotate anticlockwise.
5. Find the angular speed of the lamina at the instant when \(O B\) first becomes horizontal.
   \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-4_709_752_267_699} A smooth circular wire, with centre \(O\) and radius \(a\), is fixed in a vertical plane, and the point \(A\) is on the wire at the same horizontal level as \(O\). A small bead \(B\) of mass \(m\) can move freely on the wire. A light elastic string, with natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\), passes through a fixed ring at \(A\), and has one end fixed at \(O\) and the other end attached to \(B\). The section \(A B\) of the string is at an angle \(\theta\) above the horizontal, where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), so that \(O B\) is at an angle \(2 \theta\) to the horizontal (see diagram).
6. Taking \(O\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a ( \sqrt { 3 } + \sqrt { 3 } \cos 2 \theta + \sin 2 \theta ) .$$
7. Find the two values of \(\theta\) for which the system is in equilibrium.
8. For each position of equilibrium, determine whether it is stable or unstable.
   \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-5_478_1403_267_372} A thin horizontal rail is fixed at a height of 0.6 m above horizontal ground. A non-uniform straight \(\operatorname { rod } A B\) has mass 6 kg and length 3 m ; its centre of mass \(G\) is 2 m from \(A\) and 1 m from \(B\), and its moment of inertia about a perpendicular axis through its mid-point \(M\) is \(4.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The rod is placed in a vertical plane perpendicular to the rail, with \(A\) on the ground and \(M\) in contact with the rail. It is released from rest in this position, and begins to rotate about \(M\), without slipping on the rail. When the angle between \(A B\) and the upward vertical is \(\theta\) radians, the rod has angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\), the frictional force in the direction \(A B\) is \(F \mathrm {~N}\), and the normal reaction is \(R \mathrm {~N}\) (see diagram).
9. Show that \(\omega ^ { 2 } = 4.8 - 12 \cos \theta\).
10. Find the angular acceleration of the rod in terms of \(\theta\).
11. Show that \(F = 94.8 \cos \theta - 14.4\), and find \(R\) in terms of \(\theta\).
12. Given that the coefficient of friction between the rod and the rail is 0.9 , show that the rod will slip on the rail before \(B\) hits the ground.