4. A rough disc rotates in a horizontal plane with constant angular velocity \(\omega\) about a fixed vertical axis. A particle \(P\) of mass \(m\) lies on the disc at a distance \(\frac { 4 } { 3 } a\) from the axis. The coefficient of friction between \(P\) and the disc is \(\frac { 3 } { 5 }\). Given that \(P\) remains at rest relative to the disc,

1. prove that \(\omega ^ { 2 } \leqslant \frac { 9 g } { 20 a }\). The particle is now connected to the axis by a horizontal light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The disc again rotates with constant angular velocity \(\omega\) about the axis and \(P\) remains at rest relative to the disc at a distance \(\frac { 4 } { 3 } a\) from the axis.
2. Find the greatest and least possible values of \(\omega ^ { 2 }\).