If the horizontal and vertical forces exerted on $S$ by the cylinder are equal in magnitude, calculate the speed of $S$.
$S$ is now connected to the centre of the base of the cylinder by a horizontal light elastic string with natural length $0.25\,\text{m}$ and modulus of elasticity $13\,\text{N}$. The sphere $S$ is then set moving and describes a horizontal circle with constant angular speed $\omega\,\text{rad s}^{-1}$, while remaining in contact with both the plane base and the curved surface of the cylinder. It is given that the magnitudes of the horizontal and vertical forces exerted on $S$ by the cylinder are equal when $\omega = 8$. Calculate $m$.
Using the value of $m$ found in part (ii), determine the smallest possible value of $\omega$ for the motion.