Doppler effect observed on pendulums

Question

I wanted to investigate the doppler effect on pendulums and set up an experiment where:

• The bob of a pendulum is a speaker which emits a specific sound frequency.
• Directly underneath the lowest point of the pendulum sits a microphone that observes the frequency being emitted by the pendulum.

In the experiment, I would release the speaker from various amplitudes and record the frequency that the microphone perceives as the speaker passes the equilibrium position.

I know that the standard doppler equation is:

$$f=f_0\left(\frac{c}{v\pm c}\right)$$

where $f$ is the observed frequency

$f_0$ is the initial frequency

$c$ is the speed of sound

and $v$ is the speed of the moving source

However this equation only applies to non-accelerating systems; even though the speed of the speaker at the bottom of the pendulum can be predicted with the equation

$$v=\sqrt{2gh}$$

substituting this velocity into the doppler equation will not give accurate results because of the acceleration of the pendulum.

So, my question is, how can one calculate the frequency observed by the microphone at the bottom of the pendulum given the various release heights that the speaker is released from?

Answer 1

Your formula will give the velocity at any height on the swing down, and that can be used in the Doppler equation. Recording the sound is not a problem, but measuring the (variable) frequency may be a challenge. If you can freeze (or video) the display on an oscilloscope, you might measure the period of each cycle of the sound wave.