5.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-008_531_691_299_657}
\end{figure}
One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.
- Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
- Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\).
Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
- find, in terms of \(m\) and \(g\), the tension in the string.