# What does resonant frequency in the Q factor mean?

For the Q factor of a body undergoing force oscillations, does resonant frequency refers to the frequency of driving frequency or the body's natural frequency?

The term resonant frequency seems to mean the body's natural frequency (since this frequency corresponds to resonance); but I've just seen a question on my physics textbook of a pendulum oscillating at a frequency, f, under forced vibration and the solution used that frequency as the 'resonant frequency' in the Q factor equation to calculate the Q factor of the system.

## 2 Answers

The Q factor and the resonant frequency $$f_r$$ are properties of the system, so they don't depend on the frequency of the driving force. For a simple mass($$m$$)-spring($$k$$)-damper($$\gamma$$) system it holds that $$f_r=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$$ and $$Q=\frac{2\pi f_r}{\gamma}$$ Probably not everybody agree on the definitions and the names, but for a system with damping the resonant frequency is the same as the natural frequency of the corresponding system without damping ($$\gamma = 0$$), i.e. the expression of $$f_r$$ above. On the contrary, when damping is considered ($$\gamma\neq 0$$) the natural frequency is the frequency of the damped vibrations, i.e. $$f_n=f_r\sqrt{1-\frac{1}{4Q^2}}$$ This frequency is sometimes called "damped resonant frequency".

As far as the pendulum is concerned, I agree with niel that it was supposed that the pendulum was driven at its resonant frequency.

In your question, you state that the pendulum is oscillating at a frequency f and that it is being forced by a source. In this context it appears that what the author meant was that the pendulum was being driven at its natural frequency f.