Period of a metal rod oscillating in a magnetic field

[greg_rack]

Homework Statement

A metal rod of length $L=20cm$ and mass $m=80g$ is hung with two conductive threads of length $l=20cm$ and set in a magnetic field of magnitude $B=700mT$ directed upwards.
When the rod carries a current of $I=2.8A$ it balances itself forming an angle $\theta$ with the vertical direction.
It the rod gets moved a little from its resting position, it will oscillate with a harmonic motion around that position
$\triangleright$ Determine the period of those oscillations.

Relevant Equations

$T=\frac{1}{f}$
Harmonic motion laws
Lorentz's force equation

This problem honestly got me in big confusion.
I managed to find the angle $\theta$ at which the rod rests by equalling the components of weight and Lorentz's force... but from this point on I really don't know how to manage the harmonic oscillation part.

 

[etotheipi]

Consider an axis parallel to the rod, which passes through the two points at which the conductive threads are attached to the ceiling (or whatever).

What is the torque on the rod due to i) the two tension forces ii) the weight and iii) the magnetic force about this axis, when the rod is displaced by angle $\varepsilon$ from the equilibrium angle $\theta_0$? What is the moment of inertia of the rod about this axis? [N.B. if you like, you can write the equation of motion in terms of $\theta$, and then let $\theta = \theta_0 + \varepsilon$]

Alternatively, you could think about defining an 'effective gravitational force', and tilting your head a little bit... that's a sneaker way

 

[greg_rack]

Consider an axis parallel to the rod, which passes through the two points at which the conductive threads are attached to the ceiling (or whatever).

What is the torque on the rod due to i) the two tension forces ii) the weight and iii) the magnetic force about this axis, when the rod is displaced by angle $\varepsilon$ from the equilibrium angle $\theta_0$? What is the moment of inertia of the rod about this axis? [N.B. if you like, you can write the equation of motion in terms of $\theta$, and then let $\theta = \theta_0 + \varepsilon$]

Alternatively, you could think about defining an 'effective gravitational force', and tilting your head a little bit... that's a sneaker way

I'm sorry but it's really a lot of time since I don't work with the moment of inertia... how could I relate that to torque and finally the period?

 

[etotheipi]

There's a long way and a short way to solve the problem (let's assume we can neglect any possible variation in current due to the change in flux linked by the configuration).

You can either write the $\tau = I\ddot{\theta}$ equation, namely$$-mgl\sin{\theta} + BILl \cos{\theta} = ml^2 \ddot{\theta}$$And then let $\theta = \theta_0 + \varepsilon$, where $\varepsilon$ is the (small) angular displacement from the equilibrium position. Then, you can show using the double angle formula that $\sin{(\theta_0 + \varepsilon)} \approx \sin{\theta_0} + \varepsilon \cos{\theta_0}$ as well as $\cos{(\theta_0 + \varepsilon)} = \cos{\theta_0} - \varepsilon \sin{\theta_0}$, and if you plug this in, and use that $\ddot{\theta} = \ddot{\varepsilon}$, you'd find you end up with something in the SHM form.

However, there's a better way to solve the problem! Notice that a constant force of magnitude $\sqrt{(mg)^2 + (BIL)^2}$ acts on the bar, in a constant direction. If you tilt your head, then this is entirely equivalent to a simple pendulum swinging under the influence of an 'effective' gravitational acceleration $\sqrt{g^2 + ((BIL)/m)^2}$! Does that suggest an easier solution?

 

[greg_rack]

Consider an axis parallel to the rod, which passes through the two points at which the conductive threads are attached to the ceiling (or whatever).

What is the torque on the rod due to i) the two tension forces ii) the weight and iii) the magnetic force about this axis, when the rod is displaced by angle $\varepsilon$ from the equilibrium angle $\theta_0$? What is the moment of inertia of the rod about this axis? [N.B. if you like, you can write the equation of motion in terms of $\theta$, and then let $\theta = \theta_0 + \varepsilon$]

Alternatively, you could think about defining an 'effective gravitational force', and tilting your head a little bit... that's a sneaker way

I'm sorry for such a long delay in the response :)
Actually, I've opted for the easier way, considering the effective g on the rod and calculating the period such as a pendulum one!