Question

A flagpole consists of a flexible 4.09 m tall fiberglass pole planted in concrete. The bottom end of the flagpole is fixed in position but the top end of the flagpole is free to move. What is the lowest frequency standing wave that can be formed on the flagpole if the wave propagation speed in the fiberglass is 2730 m/s? Suppose that a standing wave on the flagpole gives rise to a sound of the same frequency. A person would be able to hear the sound produced by the lowest frequency standing wave, previously calculated, because the average human being can detect sounds at frequencies between 20 Hz and 20 kHz. A nearby mouse, however, can only detect frequencies between 1.01 Hz and 90 kHz. What is the lowest flagpole harmonic that the mouse can hear?

Answer 1

The lowest frequency standing wave that can form on a 4.09 m tall flagpole with wave speed 2730 m/s is 334.23 Hz, which is audible both to humans and mice, falling within their respective hearing ranges.

Step-by-step explanation:

The lowest frequency standing wave that can form on a 4.09 m tall fiberglass flagpole, with one end fixed, is analogous to the fundamental frequency of a vibrating string with one end fixed and one end free. This fundamental frequency (first harmonic) is given by f = v / (2L), where v is the wave speed and L is the length of the string or pole.

In this case, the wave speed v in the fiberglass is 2730 m/s, and the length L is 4.09 m. Thus, the lowest frequency f is calculated as:

f = 2730 m/s / (2 * 4.09 m) = 334.23 Hz

Since the average human can detect sounds between 20 Hz and 20 kHz, a human would be able to hear the sound produced by this standing wave. For the mouse, which can hear frequencies between 1.01 Hz and 90 kHz, it can hear this frequency as well since 334.23 Hz falls within its hearing range.