In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 550 N to 520.00 N, what beat frequency is heard when the hammer strikes the two strings simultaneously?
To find the beat frequency when the hammer strikes two strings simultaneously, we need to determine the difference in frequency between the two strings.
1. Calculate the original frequency of the string:
- Frequency (f) is directly proportional to the square root of tension (T) when length, mass, and other factors remain constant. The formula is given by:
f = k√T
Here, k is a constant.
- Since the note at 110 Hz has two strings, let's assume each string has a frequency of 110 Hz.
f1 = f2 = 110 Hz
- Now, we can rearrange the formula to find the tension of the initial string:
T = (f^2) / k^2
- Since two strings have the same frequency, we can set them equal:
(f1^2) / k^2 = (f2^2) / k^2
- Simplifying, we get:
T = T1 = T2
So, the original tension is the same for both strings.
2. Calculate the new frequency of the string with slipped tension:
- We know that the tension of one string slipped from 550 N to 520.00 N.
T2 = 520.00 N
- Now, we can use the formula to find the frequency of the string:
f2 = k√T2
3. Calculate the beat frequency:
- The beat frequency is the absolute difference between the frequencies of the two strings when struck simultaneously.
Beat Frequency (f_beat) = |f1 - f2|
- Plugging in the values:
f_beat = |f1 - f2| = |110 Hz - f2|
Now, compute the values to find the beat frequency.