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.

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 rad s\(^{-1}\) to 25 rad s\(^{-1}\).

1. Find the moment of the couple about the axis. [5]

When the flywheel is rotating with angular speed 25 rad 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 rad s\(^{-1}\).

1. Find the moment of inertia of the second flywheel about the axis. [3]