Two identical mandolin strings under 200 N of tension are sounding tones with frequencies of 577 Hz. The peg of one string slips slightly, and the tension in it drops to 190 N. How many beats per second are heard?
To find the number of beats per second, we need to understand the concept of beats in music. Beats occur when two frequencies are slightly different and interfere with each other. Beats are perceived as a fluctuation in the loudness of sound.
Given that the mandolin strings are identical, initially under a tension of 200 N, and sounding tones at a frequency of 577 Hz, we can calculate the frequency of each string.
Let's assume the frequency of the first string, which remains at 200 N of tension, is f1. The frequency of the second string, after the peg slips and the tension drops to 190 N, is f2.
Since the tension and length of the strings are the same, we can use the following formula to calculate the frequency:
f = (1/2L) √(T/μ)
Where:
- f represents the frequency
- L is the length of the string
- T is the tension in the string
- μ is the linear mass density of the string
Since the strings are identical, the linear mass density (μ) is the same for both strings. Therefore, we can simplify the formula as:
f1 = (1/2L) √(200/μ)
f2 = (1/2L) √(190/μ)
Now, we need to calculate the difference in frequency between the two strings, which will give us the number of beats per second.
Δf = |f1 - f2|
Finally, we find the number of beats per second by dividing the difference in frequency by the period:
Number of beats per second = Δf / Period
Note: The period can be calculated as the reciprocal of the frequency.
Let's calculate the number of beats per second step by step:
1. Substitute the given values into the formulas to find the frequencies of the two strings.
2. Calculate the difference in frequency between the two strings.
3. Determine the period by finding the reciprocal of the average frequency of the two strings.
4. Calculate the number of beats per second by dividing the difference in frequency by the period.
This process will give us the solution to the problem. Now, let's calculate it step by step.