7 A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\).
- Prove by integration that the moment of inertia of the rod about an axis through \(P\) perpendicular to \(A B\) is \(\frac { 4 } { 9 } m a ^ { 2 }\).
The axis through \(P\) is fixed and horizontal, and the rod can rotate without resistance in a vertical plane about this axis. The rod is released from rest in a horizontal position. Find, in terms of \(m\) and \(g\),
- the force acting on the rod at \(P\) immediately after the release of the rod,
- the force acting on the rod at \(P\) at an instant in the subsequent motion when \(B\) is vertically below \(P\).