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\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length \(a\) and modulus of elasticity \(2 m g\), are attached to a particle \(P\) of mass \(m\). The strings join the particle to the points \(A\) and \(B\) which are fixed and at a distance \(4 a\) apart on a smooth horizontal surface. The particle is at rest at the mid-point \(O\) of \(A B\). The particle is now displaced a small distance in a direction perpendicular to \(A B\), on the surface, and released from rest. At time \(t\), the displacement of \(P\) from \(O\) is \(x\) (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that \(\frac { x } { a }\) is so small that \(\left( \frac { x } { a } \right) ^ { 2 }\) and higher powers may be neglected, show that the motion of \(P\) is approximately simple harmonic and state the period of the motion.