7
\begin{figure}[h]
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\caption{Fig. 7}
\end{figure}
A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6 a\) and modulus of elasticity 3 mg . The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A , as shown in Fig. 7.
P is now pulled a further distance \(2 a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.
- Show that, until the string slackens, \(x\) satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + \frac { g x } { 2 a } = 0$$
- Determine, in terms of \(a\) and \(g\), the time at which the string slackens.
- Find, in terms of \(a\) and \(g\), the speed of P when the string slackens.