Conservation of angular momentum

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.

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.

1 Two flywheels \(F\) and \(G\) are rotating freely, about the same axis and in the same direction, with angular speeds \(21 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(36 \mathrm { rad } \mathrm { s } ^ { - 1 }\) respectively. The flywheels come into contact briefly, and immediately afterwards the angular speeds of \(F\) and \(G\) are \(28 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(34 \mathrm { rad } \mathrm { s } ^ { - 1 }\), respectively, in the same direction. Given that the moment of inertia of \(F\) about the axis is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), find the moment of inertia of \(G\) about the axis.

3 A circular disc is rotating in a horizontal plane with angular speed \(16 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis passing through its centre \(O\). The moment of inertia of the disc about the axis is \(0.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A particle, initially at rest just above the surface of the disc, drops onto the disc and sticks to it at a point 0.4 m from \(O\). Afterwards, the angular speed of the disc with the particle attached is \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle.
2. Find the loss of kinetic energy.

5 A uniform rod \(A B\) has mass \(2 m\) and length 4a.

1. Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac { 32 } { 3 } \mathrm { ma } ^ { 2 }\) The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.
2. Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\). At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.
3. Show that \(\dot { \theta } ^ { 2 } = \mathrm { k } \frac { \mathrm { g } } { \mathrm { a } } ( \cos \theta - 1 ) + \frac { 9 \mathrm { v } ^ { 2 } } { 400 \mathrm { a } ^ { 2 } }\), stating the value of the constant \(k\).
4. Find, in terms of \(a\) and \(g\), the set of values of \(v ^ { 2 }\) for which \(Q\) makes complete revolutions. When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).
5. Find \(R\) in terms of \(m\) and \(g\).
   \includegraphics[max width=\textwidth, alt={}, center]{27b790da-800f-4f5e-8f63-d52159efb48e-6_844_509_248_778} A compound pendulum consists of a uniform rod \(A B\) of length 1 m and mass 3 kg , a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac { 1 } { 3 } \mathrm {~m}\), mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(\mathrm { AP } = \mathrm { xm }\) and \(\mathrm { x } < \frac { 1 } { 2 }\) (see diagram).
6. Show that the moment of inertia of the pendulum about the axis of rotation is \(\left( 10 x ^ { 2 } - 19 x + 12 \right) \mathrm { kg } \mathrm { m } ^ { 2 }\). The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.
7. Find the angular acceleration of the pendulum, in terms of \(x , g\) and \(\theta\).
8. Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2 \pi \sqrt { \frac { 20 x ^ { 2 } - 38 x + 24 } { ( 19 - 20 x ) g } }\).
9. Hence find the value of \(x\) for which the approximate period of oscillations is least.