6 A smooth hemispherical shell of radius \(r \mathrm {~m}\) is held with its circular rim horizontal and uppermost. The centre of the rim is at the point \(O\) and the point on the inner surface directly below \(O\) is \(A\). A small object \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the inner surface of the shell so that \(\angle \mathrm { POA } = \frac { 1 } { 3 } \pi\) radians. At the instant that \(P\) is released, an impulse is applied to \(P\) in the direction of the tangent to the surface at \(P\) in the vertical plane containing \(O , A\) and \(P\). The magnitude of the impulse is denoted by \(I\) Ns.
\(P\) immediately starts to move along the surface towards \(A\) (see diagram).
\(X\) is a point on the circular rim. \(P\) leaves the shell at \(X\).
\includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-5_512_860_829_242} In an initial model of the motion of \(P\) it is assumed that \(P\) experiences no resistance to its motion.

1. Find in terms of \(r , g , m\) and \(I\) an expression for the speed of \(P\) at the instant that it leaves the shell at \(X\).
2. Find in terms of \(r , g , m\) and \(I\) an expression for the maximum height attained by \(P\) above \(X\) after it has left the shell.
3. Find an expression for the maximum mass of \(P\) for which \(P\) still leaves the shell. In a revised model it is assumed that \(P\) experiences a resistive force of constant magnitude \(R\) while it is moving.
4. Show that, in order for \(P\) to still leave the shell at \(X\) under the revised model, $$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
5. Show that the inequality from part (d) is dimensionally consistent.