OCR M4 (Mechanics 4) 2004 June

1 Two flywheels \(P\) and \(Q\) are rotating, in opposite directions, about the same fixed axis. The angular speed of \(P\) is \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the angular speed of \(Q\) is \(30 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The flywheels lock together, and after this they both rotate with angular speed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in the direction in which \(P\) was originally rotating. The moment of inertia of \(P\) about the axis is \(0.64 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). Find the moment of inertia of \(Q\) about the axis.

2 A uniform rectangular lamina has mass \(m\) and sides of length \(3 a\) and \(4 a\), and rotates freely about a fixed horizontal axis. The axis is perpendicular to the lamina and passes through a corner. The lamina makes small oscillations in its own plane, as a compound pendulum.

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

3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.

4 A uniform solid sphere, of mass 14 kg and radius 0.25 m , is rotating about a fixed axis which is a diameter of the sphere. A couple of constant moment 4.2 Nm about the axis, acting in the direction of rotation, is applied to the sphere.

1. Find the angular acceleration of the sphere. During a time interval of 30 seconds the sphere rotates through 7500 radians.
2. Find the angular speed of the sphere at the start of the time interval.
3. Find the angular speed of the sphere at the end of the time interval.
4. Find the work done by the couple during the time interval.

5 Two aircraft \(A\) and \(B\) are flying horizontally at the same height. \(A\) has constant velocity \(240 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\), and \(B\) has constant velocity \(185 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(310 ^ { \circ }\).

1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\). Initially \(A\) is 4500 m due west of \(B\). For the instant during the subsequent motion when \(A\) and \(B\) are closest together, find
2. the distance between \(A\) and \(B\),
3. the bearing of \(A\) from \(B\).

6
\includegraphics[max width=\textwidth, alt={}, center]{fb9e4e4a-953b-4e52-858e-438b4009e79c-3_428_595_221_806} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(m g\); one end is attached to \(B\) and the other end is attached to a light ring \(R\) which can slide along a smooth vertical wire. The wire is in the same vertical plane as \(A B\), and is at a distance \(a\) from \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical, where \(0 < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Give a reason why the string \(R B\) is always horizontal.
2. By considering potential energy, find the value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

7 A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\).

1. Prove by integration that the moment of inertia of the rod about an axis through \(P\) perpendicular to \(A B\) is \(\frac { 4 } { 9 } m a ^ { 2 }\). The axis through \(P\) is fixed and horizontal, and the rod can rotate without resistance in a vertical plane about this axis. The rod is released from rest in a horizontal position. Find, in terms of \(m\) and \(g\),
2. the force acting on the rod at \(P\) immediately after the release of the rod,
3. the force acting on the rod at \(P\) at an instant in the subsequent motion when \(B\) is vertically below \(P\).