7 A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity \(4 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(A\). The particle is released from rest at a point 4.8 m vertically below \(A\). At time \(t \mathrm {~s}\) after \(P\) 's release \(P\) is ( \(4 + x ) \mathrm { m }\) below \(A\).

1. Show that \(4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x\).
   \(P\) 's motion is simple harmonic.
2. Write down the amplitude of \(P\) 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle \(Q\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(B\). The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time \(t \mathrm {~s}\) after \(Q\) 's release \(B Q\) makes an angle of \(\theta\) radians with the downward vertical.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta\). The period of the simple harmonic motion to which \(Q\) 's motion approximates is the same as the period of \(P\) 's motion.
4. Given that \(\theta = 0.08\) when \(t = 0\), find the speed of \(Q\) when \(t = 0.25\).