7. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of modulus \(m g \mathrm {~N}\) and natural length \(l \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(O X = \frac { 5 l } { 4 } \mathrm {~m}\).

1. Find the coefficient of friction between \(P\) and the table.
   \(P\) is now given a small displacement \(x \mathrm {~m}\) horizontally along \(O X\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
2. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. If \(P\) is held at the point where the extension in the string is \(l m\) and then released,
3. show that the string becomes slack after a time \(\left( \frac { \pi } { 2 } + \arcsin \left( \frac { 1 } { 3 } \right) \right) \sqrt { \frac { l } { g } } \mathrm {~s}\).
4. Determine the speed of \(P\) when it reaches \(O\).