A train moving with a speed of 31.8 m/s sounds a 136-Hz horn. Suppose a stationary observer sounds a horn that is identical to the one on the train. What frequency is heard from this horn by a passenger in the train.
a) 9.8 Hz
b) 31.8 Hz
c) 99 Hz
d) 115 Hz
e) 149 Hz
The fundamental of a closed organ pipe is 261.6 Hz (middle C). The second harmonic of an open organ pipe has the same frequency
what are the lengths of the closed pipe?
A)1m b) .54m C).48 m d) .33m
What are the lengths of open pipe?
A)1.31m b)1m c).37m d).24m
First question: It depends on the relative speed the passenger has...is he going away, or coming toward the sound?
On the second question, the length of the pipe if closed on one end, is lambda/4, or length= 1/4 340/261= you do it.
The second harmonic has a length lambda, so l= 340/261= you do it.
To solve the first problem, we can use the formula for the Doppler effect. The formula for the observed frequency (f') is:
f' = f * ((v + vo) / (v + vs))
where f is the frequency emitted by the source (136 Hz), v is the speed of sound (approximately 343 m/s), vo is the velocity of the observer (0 m/s because the observer is stationary), and vs is the velocity of the source (31.8 m/s).
Plugging in the values, we get:
f' = 136 * ((343 + 0) / (343 + 31.8))
= 136 * (343 / 374.8)
= 125.3 Hz
So the frequency heard by the passenger in the train is approximately 125.3 Hz. None of the given answer choices match this value.
For the second problem, the fundamental frequency of a closed pipe is given as 261.6 Hz. Since the second harmonic of an open pipe has the same frequency, the length of an open pipe can be calculated using the formula:
L = λ / 2
where L is the length of the pipe and λ is the wavelength. For the second harmonic (n = 2), the wavelength is equal to twice the length of the pipe:
λ = 2L
Substituting this value into the previous equation, we get:
2L = λ / 2
L = λ / 4
Since the fundamental frequency is 261.6 Hz, the wavelength can be calculated using the formula for the speed of sound:
v = f * λ
where v is the speed of sound (approximately 343 m/s) and f is the frequency (261.6 Hz). Rearranging the equation, we get:
λ = v / f
Substituting the values, we get:
λ = 343 / 261.6
≈ 1.31 m
Finally, substituting this value back into the equation for the length of the open pipe:
L = 1.31 / 4
≈ 0.33 m
So the length of the closed pipe is approximately 0.33 m (answer choice d), and the length of the open pipe is approximately 1.31 m (answer choice a).
To find the frequency heard by the passenger in the train, we need to consider the Doppler effect. The formula for the apparent frequency observed by the observer is given by:
f' = f * (v + u) / (v + vobs)
Where:
f' = apparent frequency observed by the observer
f = frequency emitted by the stationary observer
v = speed of sound
u = speed of the train
vobs = speed of the observer (which is 0 since the observer is stationary)
In this case, the frequency emitted by the stationary observer is 136 Hz, and the speed of sound is approximately 343 m/s. The speed of the train is given as 31.8 m/s.
By substituting the values into the formula, we get:
f' = 136 * (343 + 31.8) / (343 + 0)
f' = 136 * 374.8 / 343
f' ≈ 148.83 Hz
Therefore, the frequency heard from the horn by the passenger in the train is approximately 148.83 Hz. Option e) 149 Hz is the closest answer.
For the second question about the lengths of the closed and open organ pipes, we can use the formula for the length of a closed pipe (fundamental frequency) and an open pipe (second harmonic):
Length of a closed pipe (fundamental): L = (v / 4) / f
Length of an open pipe (second harmonic): L = (v / 2) / f
Given that the fundamental frequency is 261.6 Hz (middle C) and the speed of sound is approximately 343 m/s, we can calculate the length of the closed pipe:
L = (343 / 4) / 261.6
L ≈ 0.413 m
Therefore, the length of the closed pipe is approximately 0.413 m. None of the options provided match this value.
For the length of the open pipe (second harmonic), we use the same formula but with the frequency doubled:
L = (343 / 2) / (2 * 261.6)
L ≈ 0.328 m
So, the length of the open pipe is approximately 0.328 m. The closest option is d) 0.33 m.