The overall length of a piccolo is 30.6 cm. The resonating air column vibrates as in a pipe that is open at both ends. Assume the speed of sound is 343 m/s. (a) Find the frequency of the lowest note a piccolo can play, assuming the speed of sound in air is 343 m/s. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 41.0 kHz, find the distance between adjacent antinodes for this mode of vibration. Assume the speed of sound is 343 m/s.

Question

The overall length of a piccolo is 30.6 cm. The resonating air column vibrates as in a pipe that is open at both ends. Assume the speed of sound is 343 m/s. (a) Find the frequency of the lowest note a piccolo can play, assuming the speed of sound in air is 343 m/s. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 41.0 kHz, find the distance between adjacent antinodes for this mode of vibration. Assume the speed of sound is 343 m/s.

Answer 1

To find the frequency of the lowest note a piccolo can play, we need to determine the wavelength of the sound wave produced. Since the pipe is open at both ends, the fundamental frequency (lowest note) will have a wavelength twice the length of the piccolo.

Step 1: Convert the length of the piccolo to meters:
Length = 30.6 cm = 0.306 m

Step 2: Calculate the wavelength:
Wavelength = 2 × Length
= 2 × 0.306 m
= 0.612 m

Step 3: Use the formula for the speed of sound to find the frequency:
Speed of sound = Frequency × Wavelength

Rearranging the formula to solve for the frequency, we have:
Frequency = Speed of sound / Wavelength

Substituting the given values:
Frequency = 343 m/s / 0.612 m ≈ 561.44 Hz

Therefore, the frequency of the lowest note a piccolo can play is approximately 561.44 Hz.

For part (b), to find the distance between adjacent antinodes for the highest note a piccolo can sound, we need to calculate the wavelength of the sound wave produced. Since the pipe length changes due to the opened side holes, we need to consider the effective length of the resonant column.

Step 1: Find the effective length of the resonant column:
Length = Speed of sound / Frequency
Length = 343 m/s / (41,000 Hz × 2)
Length ≈ 4.18 cm

Step 2: Calculate the wavelength:
Wavelength = 2 × Length
= 2 × 0.0418 m
= 0.0836 m

Step 3: Use the formula for the speed of sound to find the distance between adjacent antinodes:
Distance = Wavelength / 2
= 0.0836 m / 2
= 0.0418 m

Therefore, the distance between adjacent antinodes for this mode of vibration in the highest note a piccolo can sound is approximately 0.0418 meters.