7.
\begin{figure}[h]
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\caption{Figure 5}
\includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-7_492_929_504_543}
\end{figure}
A small ring \(R\) of mass in is free to slide on a smooth straight wire which is fixed at an angle of \(30 ^ { \circ }\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(A B = \frac { 9 } { 8 } a\).
- Show that \(\lambda = 4 m g\).
The ring is pulled down to the point \(C\), where \(B C = \frac { 1 } { 4 } a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is ( \(\frac { 1 } { 8 } a + x\) ).
- Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi \sqrt { \left( \frac { a } { g } \right) }\).
(6) - Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut.
(2) - Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time.
END