4. The diagram shows three light rods \(A B , B C\) and \(C A\) rigidly joined together so that \(A B C\) is a right-angled triangle with \(A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}\) and \(\widehat { A B } = 90 ^ { \circ }\). The rods support a uniform lamina, of density \(2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), in the shape of a quarter circle \(A D E\) with radius 12 cm and centre at the vertex \(A\). Three particles are attached to \(B C\) : one at \(B\), one at \(C\) and one at \(F\), the midpoint of \(B C\). The masses at \(C , F\) and \(B\) are \(50 m \mathrm {~kg} , 30 m \mathrm {~kg}\) and \(20 m \mathrm {~kg}\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}

1. Calculate the distance of the centre of mass of the system from
   1. \(A C\),
   2. \(A B\).
2. When the system is freely suspended from a point \(P\) on \(A C\), it hangs in equilibrium with \(A B\) vertical. Write down the length \(A P\).
3. When the system is freely suspended from a point \(Q\) on \(A D\), it hangs in equilibrium with \(Q B\) making an angle of \(60 ^ { \circ }\) with the vertical. Find the distance \(A Q\).