Rotating disc/platform system

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\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-4_673_773_269_685} A horizontal circular disc of radius 4 m is free to rotate about a vertical axis through its centre \(O\). One end of a light inextensible rope of length 5 m is attached to a point \(A\) of the circumference of the disc, and an object \(P\) of mass 24 kg is attached to the other end of the rope. When the disc rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\), the rope makes an angle of \(\theta\) radians with the vertical and the tension in the rope is \(T \mathrm {~N}\) (see diagram). You may assume that the rope is always in the same vertical plane as the radius \(O A\) of the disc.

1. Given that \(\cos \theta = \frac { 24 } { 25 }\), find the value of \(\omega\).
2. Given instead that the speed of \(P\) is twice the speed of the point \(A\), find
   (a) the value of \(T\),
   (b) the speed of \(P\).

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\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis through its centre \(O\). One end of a light inextensible string of length 0.8 m is attached to a point \(A\) of the circumference of the disc. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius \(O A\) of the disc. The string makes a constant angle \(\theta\) with the vertical (see diagram). The speed of \(P\) is 1.6 times the speed of \(A\).

1. Show that \(\sin \theta = \frac { 3 } { 8 }\).
2. Find the tension in the string.
3. Find the value of \(\omega\).