Time period of a pendulum when a constant horizontal force acts

Question

The time period of a pendulum is given by $$T=2\pi\sqrt{\frac{l}{g}}$$

Will the time period change if a constant horizontal force acts on the pendulum? For example, if a force $F$ acts on the Bob along the horizontal. Or applying an electric field along the horizontal and giving the Bob a charge.

I think the time period won't change but the equilibrium position will change. The only restoring force is gravity. I tried to prove it:

For small displacement of $\theta$ about point of suspension, the torque about the point of suspension is $$\tau=F\cos\theta\cdot l - mg\sin\theta\cdot l$$ $$I\alpha=F\cos\theta\cdot l - mg\sin\theta\cdot l$$ For small angular displacement, $\sin\theta=\theta$ $$I\alpha=F\cos\theta\cdot l - mg\theta\cdot l$$ For SHM, $\alpha$ should be proportional to $\theta$. The only term that obeys the SHM rule is $mgl\theta$.

Answer 1

Gravity exerts a constant force downwards. If you apply a second constant force sideways you are in effect rotating the gravitational force:

where the angle $\theta$ is given by:

$$ \tan\theta = \frac{F_\text{ext}}{F_g} $$

The modulus of the net force is given by the usual Pythagoras rule:

$$ F_\text{net}^2 = F_g^2 + F_\text{ext}^2 $$

So if you imagine mentally rotating the whole experiment an angle $\theta$ anticlockwise you would have the pendulum handing vertically downwards in an effective gravitational field of $F_\text{net}$. This should make it obvious how the period changes.