7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to points \(C\) and \(D\) on the same horizontal level by means of two light inextensible strings \(C P\) and \(D P\), both of length \(40 \mathrm {~cm} . P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) so as to move in a vertical circle in a plane perpendicular to \(C D\), so that angle \(P C D =\) angle \(P D C = \theta\) throughout the motion.
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_335_405_1775_1572} If \(u\) is just large enough for the strings to remain taut as \(P\) describes this circular path,

1. show that \(u ^ { 2 } = 2 g \sin \theta\). The string \(D P\) breaks when \(P\) is at its lowest point. \(P\) then immediately starts to move in a horizontal circle on the end of the string \(C P\).
2. Prove that \(\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }\).