Problem

A fixed smooth spherical shell has centre $O$ and radius $a$. A particle of mass $m$ moves in complete vertical circles on the smooth inner surface of the shell, where the plane of the circular motion is vertical and passes through $O$. The particle has speed $v$ when it is at point $A$, where $OA$ makes an angle $\theta$ with the upward vertical through $O$, and $\cos\theta=\frac1{18}$ (see diagram).

(a) Show that

$$v\geq\frac13\sqrt{26ag}.$$

It is given that

$$v=\frac13\sqrt{26ag}.$$

(b) Find, in terms of $m$ and $g$, an expression for the greatest possible value of the normal reaction between the shell and the particle.