I have 2 questions I can't solve. I am Stuck.
1. A person with very good musical pitch listens to a guitarist on a moving stage play a high A (880 Hz) while the stage is moving away at 4.0 m/s on a cold day, 0 °C. What frequency does this person hear? If they can discriminate 0.25 Hz can they tell the difference between this note from a moving guitarist and a stationary guitarist?
2. Find the frequency of these modes if the air temperature 20 °C. A guitar string 1.3 m long with a wave speed of 66 m/s.
Sure! Let's break down each question and explain how to solve them:
1. To find the frequency that the person with good musical pitch hears, we need to use the Doppler effect equation. The equation is given by:
f' = f ((c + v) / (c + vs))
Where f' is the observed frequency, f is the source frequency (880 Hz in this case), c is the speed of sound at 0 °C (approximately 331 m/s), v is the velocity of the stage (4.0 m/s), and vs is the speed of sound at 0 °C.
To calculate the speed of sound at 0 °C, we can use the formula:
vs = sqrt(gamma * R * T)
Where gamma is the adiabatic index for air (approximately 1.4), R is the gas constant for air (approximately 287 J/(kg*K)), and T is the temperature in Kelvin (which is 0 °C + 273.15 = 273.15 K).
Now, substituting the values into the equation:
vs = sqrt(1.4 * 287 * 273.15) ≈ 331.3 m/s
Plugging this value into the Doppler effect equation:
f' = 880 ((331 + 4) / (331 + 331.3)) ≈ 870.4 Hz
Therefore, the person would actually hear a frequency of approximately 870.4 Hz.
Now, to determine if the person can discriminate a difference of 0.25 Hz, we need to compare the frequency difference between the moving and stationary guitarists. The frequency difference can be calculated as:
Δf = |f' - f|
Δf = |870.4 - 880| ≈ 9.6 Hz
Since the frequency difference is larger than the person's ability to discriminate (0.25 Hz), they would be able to tell the difference between the note from a moving guitarist and a stationary guitarist.
2. To find the frequency of the modes of a guitar string, we can use the formula:
f = (n * v) / (2L)
Where f is the frequency, n is the mode number (1, 2, 3, ...), v is the wave speed (66 m/s), and L is the length of the guitar string (1.3 m).
Substituting the given values into the equation:
For the fundamental frequency (n = 1):
f1 = (1 * 66) / (2 * 1.3) ≈ 25.4 Hz
For the second mode (n = 2):
f2 = (2 * 66) / (2 * 1.3) ≈ 50.8 Hz
And so on for higher modes.
Therefore, the frequencies of the modes for a guitar string with a length of 1.3 m and a wave speed of 66 m/s at a temperature of 20 °C would be approximately 25.4 Hz for the fundamental frequency and 50.8 Hz for the second mode.