One end of a light inextensible string of length l is attached to the vertex of a smooth cone of semivertical angle \(45°\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(ω\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).

\((\text{i})\) By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T=\dfrac{1}{2}m \left ( \sqrt{2} g + l \omega ^2 \right )\).

\((\text{ii})\) When the string has length \(0.8 m\), calculate the greatest value of \(ω\) for which the particle remains in contact with the cone, to 3 significant figures.

**Input \(100 \times\) your answer to part \((\text{ii})\).**

###### There are **6** marks available for part (i) and **4** marks for part (ii).

###### In total, this question is worth **13.9%** of all available marks in the paper.