Difficult# Period of a Dipole Pendulum

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The figure below depicts an electric dipole arranged symmetrically about the origin in the $xy$-plane.

The dipole moment arm makes an initial angle of $\theta_0 = 10.4^{\circ}$ with respect to the positive $y$-axis. The magnitude of each of the two point charges is $q = 32.4 \ \mu\rm{C}$, and the charges are separated by a distance $d = 22.7 \text{ mm}$ from one another. The mass of either charge is $m = 75.3 \text{ g}$.

If this dipole is placed in a uniform electric field, $\vec{E} = -634 \hat{y} \rm{\text{ N}/\text{C}}$, it will oscillate back and forth across the positive $y$-axis, about the positive $z$-axis, forming a torsional pendulum. You may assume that the initial angle of deflection, $\theta_0$, is small enough so that $\sin\theta_0 \approx \theta_0$.

What is the period $T$ of this pendulum?