3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-08_583_549_210_760}
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\caption{Figure 1}
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A light inextensible string of length \(7 l\) has one end attached to a fixed point \(A\) and the other end attached to a fixed point \(B\), where \(A\) is vertically above \(B\) and \(A B = 5\) l. A particle of mass \(m\) is attached to the string at the point \(C\) where \(A C = 4 l\), as shown in Figure 1. The particle moves in a horizontal circle with constant angular speed \(\omega\). Both parts of the string are taut.
- Find, in terms of \(m , g , l\) and \(\omega\),
- the tension in \(A C\),
- the tension in \(B C\).
The time taken by the particle to complete one revolution is \(R\).
Given that \(R \leqslant k \pi \sqrt { \frac { l } { 5 g } }\)
- find the least possible value of \(k\).