2 A small ring of mass \(m\) can slide freely along a fixed smooth horizontal rod. A light elastic string of natural length \(a\) and stiffness \(k\) has one end attached to a point A on the rod and the other end attached to the ring. An identical elastic string has one end attached to the ring and the other end attached to a point B which is a distance \(a\) vertically above the rod and a horizontal distance \(2 a\) from the point A . The displacement of the ring from the vertical line through B is \(x\), as shown in Fig. 2.
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\includegraphics[alt={},max width=\textwidth]{0166dd50-5069-47f4-a015-d01a9c54faf4-2_405_1063_1270_539}
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\caption{Fig. 2}
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- Find an expression for \(V\), the potential energy of the system when \(0 < x < a\), and show that
$$\frac { \mathrm { d } V } { \mathrm {~d} x } = 2 k x - k a - \frac { k a x } { \sqrt { a ^ { 2 } + x ^ { 2 } } }$$
- Show that \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) is always positive.
- Show that there is a position of equilibrium with \(\frac { 1 } { 2 } a < x < a\). State, with a reason, whether it is stable or unstable.