Simple Pendulum SHM Problem: Calculating Angular Frequency for Maximum Amplitude

[Hernaner28]

Homework Statement

Consider, as shown in the picture, a penulum of mass m=0.52g, hanging on an ideal rope of length l=0.31m. A force is exerted to this pendulum as shown in the picture.
The horizontal component of the force is:
$$\displaystyle {{F}_{x}}={{F}_{0}}\cos \left( \omega t \right)$$
$$\displaystyle {{F}_{0}}=2.3N$$

Calculate the angular frequency that will make the system oscilate with the maximum amplitud. Consider small oscilations around the equilibrium position.

Homework Equations

The Attempt at a Solution

I did nothing beucase I don't know how to deal with that variable force. BUT, what I did was to use the angular frequency of a simple pendulim which is
$$\displaystyle \sqrt{\frac{g}{L}}$$
And I just replaced the values and I got the correct answer! 5.6rad/s.
But my question is: why did I get the correct result if there's a force there?

Thanks!

 

[Infinitum]

Hi Hernaner28!

Your method works because the frequency you used is the natural frequency of the system. And anybody vibrates with a greater amplitude when externally forced to vibrate with its natural frequency(Resonance)