3. \begin{figure}[h] \end{figure} A light inextensible string has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). An identical string has one end attached to the fixed point \(B\), where \(B\) is vertically below \(A\) and \(A B = 4 a\), and the other end attached to \(P\), as shown in Figure 2. The particle is moving in a horizontal circle with constant angular speed \(\omega\), with both strings taut and inclined at \(30 ^ { \circ }\) to the vertical. The tension in the upper string is twice the tension in the lower string. Find \(\omega\) in terms of \(a\) and \(g\).