Centre of mass of rotating body

1 A uniform lamina is in the form of a sector of a circle with centre \(O\), radius 0.2 m and angle 1.5 radians. The lamina rotates in a horizontal plane about a fixed vertical axis through \(O\). The centre of mass of the lamina moves with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the angular speed of the lamina is \(3.30 \mathrm { rad } \mathrm { s } ^ { - 1 }\), correct to 3 significant figures.

1 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_373_552_260_799} A uniform isosceles triangular lamina \(A B C\) is right-angled at \(B\). The length of \(A C\) is 24 cm . The lamina rotates in a horizontal plane, about a vertical axis through the mid-point of \(A C\), with angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Find the speed with which the centre of mass of the lamina is moving.
[0pt] [3]

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

A uniform circular pulley, of mass \(4m\) and radius \(r\), is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and has a particle of mass \(2m\) attached to one end and a particle of mass \(3m\) attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.

1. Find the angular acceleration of the pulley. [8]

When the angular speed of the pulley is \(\Omega\), the string breaks and a constant braking couple of magnitude \(G\) is applied to the pulley which brings it to rest.

1. Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest. [4]