3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .

1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
4. Find the period and the amplitude of the motion.
5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
6. Find the total distance travelled by P during the first 2.4 seconds of its motion.