6. A beam \(A B\) has length 15 m . The beam rests horizontally in equilibrium on two smooth supports at the points \(P\) and \(Q\), where \(A P = 2 \mathrm {~m}\) and \(Q B = 3 \mathrm {~m}\). When a child of mass 50 kg stands on the beam at \(A\), the beam remains in equilibrium and is on the point of tilting about \(P\). When the same child of mass 50 kg stands on the beam at \(B\), the beam remains in equilibrium and is on the point of tilting about \(Q\). The child is modelled as a particle and the beam is modelled as a non-uniform rod.

1. 1. Find the mass of the beam.
   2. Find the distance of the centre of mass of the beam from \(A\). When the child stands at the point \(X\) on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,
2. find \(A X\).