6. A light elastic string of natural length \(a\) has modulus of elasticity \(k m g\), where \(k\) is a constant. One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle of mass \(m\). The particle moves, with the string stretched, in a horizontal circle with constant angular speed \(\omega\), with the centre of the circle vertically below \(O\).

1. Show that, if the string makes a constant angle \(\theta\) with the vertical, $$\cos \theta = \frac { k g - a \omega ^ { 2 } } { k a \omega ^ { 2 } }$$
2. Show that \(\omega < \sqrt { \frac { k g } { a } }\)
   [0pt] [Question 6 Continued] Spare space for extra working Spare space for extra working Spare space for extra working Spare space for extra working
   [0pt] [End of Examination]