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 \(X\).
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In an initial model of the motion of \(P\) it is assumed that \(P\) experiences no resistance to its motion.
- 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 \(X\).
- 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.
- 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.
- Show that, in order for \(P\) to still leave the shell at \(X\) under the revised model,
$$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
- Show that the inequality from part (d) is dimensionally consistent.