Two identical guitar strings are stretched with the same tension between supports that are not the same distance apart. The fundamental frequency of the higher-pitched string is 400Hz, and the speed of transverse waves in both wires is 200 m/s. How much longer is the lower-pitched string if the beat frequency is 4Hz?
To find the length difference between the two guitar strings, we can use the formula for beat frequency:
Beat frequency = |frequency1 - frequency2|
Given:
- Frequency of string 1 (higher-pitched string) = 400 Hz
- Frequency of string 2 (lower-pitched string) = ?
- Beat frequency = 4 Hz
First, let's determine the frequency of string 2. Since the strings are identical, they have the same tension and the same wave speed:
Wave speed = 200 m/s
Fundamental frequency of a string is given by the formula:
Fundamental frequency = Wave speed / (2 * Length)
For string 2:
Frequency2 = Wave speed / (2 * Length2)
Now, we need to find the length of string 2. The difference in beat frequency is caused by the difference in length between the two strings. Let's assume the length of string 2 is longer than string 1 by ΔL.
Length2 = Length1 + ΔL
Substituting this into the frequency formula for string 2:
Frequency2 = Wave speed / (2 * (Length1 + ΔL))
Now, we can calculate the difference in frequency between the two strings:
|Frequency1 - Frequency2| = Beat frequency
|400 Hz - (Wave speed / (2 * (Length1 + ΔL)))| = 4 Hz
We have an absolute value equation, so we need to consider two cases:
Case 1: Frequency1 - Frequency2 = Beat frequency
400 Hz - (Wave speed / (2 * (Length1 + ΔL))) = 4 Hz
Solving for ΔL:
ΔL = (Wave speed / 2) * (400 Hz - 4 Hz) / (400 Hz)
Case 2: Frequency2 - Frequency1 = Beat frequency
(Wave speed / (2 * (Length1 + ΔL))) - 400 Hz = 4 Hz
Solving for ΔL:
ΔL = (Wave speed / 2) * (400 Hz - 4 Hz) / (400 Hz)
In both cases, the value of ΔL will be the same.
Now, substitute the given values:
ΔL = (200 m/s / 2) * (400 Hz - 4 Hz) / (400 Hz)
ΔL = (100 m/s) * (396 Hz) / (400 Hz)
ΔL = (99,600 m/s * Hz) / (400 Hz)
ΔL = 249 m
Therefore, the lower-pitched string is 249 meters longer than the higher-pitched string.