Two strings are adjusted to vibrate at exactly 224 Hz. Then the tension in one string is increased slightly. Afterward, three beats per second are heard when the strings vibrate at the same time. What is the new frequency of the string that was tightened? (in Hz)
What is 227?
To find the new frequency of the tightened string, we need to understand the concept of beats and the relationship between beat frequency and the difference in frequencies between two vibrating objects.
Beats occur when two waves with slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the frequencies of the two waves.
In this case, when the two strings vibrate at the same time, three beats per second are heard. This means that the beat frequency is 3 Hz.
Let's denote the original frequency of one string (before tightening) as f1, and the frequency of the tightened string as f2.
The beat frequency (bf) can be calculated using the formula:
bf = |f1 - f2|
Given that bf = 3 Hz, we can rearrange the formula to:
|f1 - f2| = 3
Since we know that both strings vibrate at exactly 224 Hz initially (before tightening), we can substitute the values into the formula:
|224 - f2| = 3
To find the two possible values for f2, we set up two equations:
1) 224 - f2 = 3
2) -(224 - f2) = 3
Solving equation 1, we have:
224 - f2 = 3
f2 = 224 - 3
f2 = 221 Hz
Solving equation 2, we have:
-(224 - f2) = 3
f2 - 224 = 3
f2 = 227 Hz
Thus, the new frequency of the tightened string can either be 221 Hz or 227 Hz, depending on the direction of the tension adjustment.