4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-07_805_460_214_740}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.
- Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
- Find the tension in \(B P\).
- Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).