OCR M4 (Mechanics 4) 2003 June

1 A propeller shaft has constant angular acceleration. It turns through 160 radians as its angular speed increases from \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find

1. the angular acceleration of the propeller shaft,
2. the time taken for this increase in angular speed.

2
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_462_490_610_834} The diagram shows a uniform lamina \(A B C D E F\) in which all the corners are right angles. The mass of the lamina is \(3 m\).

1. Show that the moment of inertia of the lamina about \(A B\) is \(3 m a ^ { 2 }\).
2. Find the moment of inertia of the lamina about an axis perpendicular to the lamina and passing through \(A\).

3 A uniform rod, of mass 0.75 kg and length 1.6 m , rotates in a vertical plane about a fixed horizontal axis through one end. A frictional couple of constant moment opposes the motion. The rod is released from rest in a horizontal position and, when the rod is first vertical, its angular speed is \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the magnitude of the frictional couple.
   \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-2_584_527_1798_822} A disc is rotating about the same axis. The moment of inertia of the disc about the axis is \(0.56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When the rod is vertical, the disc has angular speed \(4.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the opposite direction to that of the rod (see diagram). At this instant the rod hits a magnetic catch \(C\) on the disc and becomes attached to the disc.
2. Find the angular speed of the rod and disc immediately after they have become attached.

4 A cruise ship \(C\) is sailing due north at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat \(B\), initially 2000 m due west of \(C\), sails with constant speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight line course which takes it as close as possible to \(C\).

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

5 The region bounded by the \(x\)-axis, the line \(x = 8\) and the curve \(y = x ^ { \frac { 1 } { 3 } }\) for \(0 \leqslant x \leqslant 8\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. The unit of length is the metre, and the density of the solid is \(350 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).

1. Show that the mass of the solid is \(6720 \pi \mathrm {~kg}\).
2. Find the \(x\)-coordinate of the centre of mass of the solid.
3. Find the moment of inertia of the solid about the \(x\)-axis.

6
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-3_468_550_1201_824} A wheel consists of a uniform circular disc, with centre \(O\), mass 0.08 kg and radius 0.35 m , with a particle \(P\) of mass 0.24 kg attached to a point on the circumference. The wheel is rotating without resistance in a vertical plane about a fixed horizontal axis through \(O\) (see diagram).

1. Find the moment of inertia of the wheel about the axis.
2. Find the distance of the centre of mass of the wheel from the axis. At an instant when \(O P\) is horizontal and the angular speed of the wheel is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
3. the angular acceleration of the wheel,
4. the magnitude of the force acting on the wheel at \(O\).

7
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-4_557_1036_278_553} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is pivoted to a fixed point at \(A\) and is free to rotate in a vertical plane. Two fixed vertical wires in this plane are a distance \(6 a\) apart and the point \(A\) is half-way between the two wires. Light smooth rings \(R _ { 1 }\) and \(R _ { 2 }\) slide on the wires and are connected to \(B\) by light elastic strings, each of natural length \(a\) and modulus of elasticity \(\frac { 1 } { 4 } m g\). The strings \(B R _ { 1 }\) and \(B R _ { 2 }\) are always horizontal and the angle between \(A B\) and the upward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a \left( 1 + \cos \theta + \sin ^ { 2 } \theta \right) .$$
2. Given that \(\theta = 0\) is a position of stable equilibrium, find the approximate period of small oscillations about this position.