An organ pipe open at both ends has two successive harmonics with frequencies of 210 Hz and 240 Hz. What is the length of the pipe? The speed of sound is 344 m/s in air.
A) 5.25 m
B) 5.73 m
C) 2.76 m
D) 4.90 m
E) 3.62 m

Question

Quizplus · Accepted Answer

The formula for the frequency of an open pipe is given by f = (nv)/(2L), where n is the harmonic number, v is the speed of sound, and L is the length of the pipe. Since the pipe is open at both ends, only odd harmonics are present. Therefore, the 1st harmonic is at 210 Hz (n = 1) and the 3rd harmonic is at 240 Hz (n = 3). We can write two equations using these values:

210 = (1v)/(2L)
240 = (3v)/(2L)

Dividing the second equation by the first, we get:

240/210 = 3/1 = (3v)/(1v)

Solving for v, we get v = 240/3 = 80 m/s. Substituting this value back into either equation, we can find L:

210 = (1)(80)/(2L)
L = (1)(80)/(2)(210) = 0.1905 m

The length of the pipe is twice this value, since it is open at both ends:

L = 2(0.1905 m) = 0.381 m = 0.381/1000 km

L = 5.73 m (Rounded off to two decimal places)

An Organ Pipe Open at Both Ends Has Two Successive