Oscillating wire

First, we find the torque due to the magnetic force , that will be equal to torque of (gravity and buoyant force) at equilibrium.

Clearly, as density of liquid is $2 \rho$ , the net force except magnetic force is $2mg - mg = mg$ upwards, creating an anticlockwise torque of $mg \frac{L}{2} \sin \theta$ about hinge.

Consider a small element of length $dx$ , at a distance $x$ from the hinge on the smaller element. The magnetic force on this element is shown.

Clearly, the magnetic field here due to long wire is given by :

$\displaystyle B = \frac{\mu_{0} I}{2 \pi x \sin \theta} \bigg(\frac{1+ \cos \theta}{2}\bigg)$ out of the plane.

The force on the element is as shown,

$\displaystyle dF = I B dx = \frac{\mu_{0} I^2}{2 \pi x \sin \theta} \big(\frac{1+ \cos \theta}{2}\big) dx$

Hence, $\displaystyle d \tau_{B}= x dF_{B} = \frac{\mu_{0} I^2 (1+ \cos \theta)}{4 \pi \sin \theta} dx$

$\Rightarrow \displaystyle d \tau_{B} = \frac{\mu_{0} I^2 \cot \frac{\theta}{2}}{4 \pi} dx$

Thus, $\displaystyle \tau_{B} = \frac{\mu_{0} I^2 L}{4 \pi} \cot \frac{\theta}{2}$

At $\theta = 60^{\circ}$ , the torques balance,

$\displaystyle \frac{\mu_{0} I^2 L}{4 \pi} \cot \frac{60^{\circ}}{2} = mg \frac{L}{2} \sin 60^{\circ}$

$\Rightarrow \frac{\mu_{0} I^2}{\pi} = mg$

Put this value in the expression of torque due to magnetic force to get :

$\displaystyle \tau_{B} = \frac{mg L}{4} \cot \frac{\theta}{2}$

Hence, $\tau_{net} = \frac{mg L}{4} \cot \frac{\theta}{2} - \frac{mgL}{2} \cos \theta$ clockwise

Now, when displaced by small angle $\phi$ , Wire would restore. Differentiate above expression to get

$\displaystyle \frac{\delta \tau_{net}}{\phi} \approx -\frac{mg L}{8} \csc^2 \frac{\theta}{2} - \frac{mgL}{2} \cos \theta$

Since, initially, $\tau_{net}= 0$ , $\delta \tau_{net} = \tau_{net}$

At $\theta = 60^{\circ}$ ,

$\delta \tau_{net} = \tau_{net} = \frac{-3mgL}{4} \phi$

Hence, $\frac{mL^2}{3} \ddot{\phi} = - \frac{3mgL}{4} \phi$

Hence, we get $\displaystyle \boxed{T = \sqrt{\frac{4L}{9g}}}$

Substitute the values to get $T \approx 1.33806 s$