How Do Tension Changes Affect the Frequency of Violin Strings?

[eku_girl83]

Well, I've encountered a couple more problems...
Here they are:
Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 196 Hz. One of the strings is retuned by adjusting its tension. When this is done, 1.5 beats per second are heard when both strings are plucked simultaneously. I found the lower and upper limits on the possible fundamental frequencies of the re-tuned strings to be 194.5 and 197.5 Hz, respectively. I'm struggling with this:
By what fractional amount was the string tension changed if it was increased?
By what fractional amount was the string tension changed if it was decreased?
(delta T)/T

Could someone help me with this? Thanks!

 

[swansont]

Well, I've encountered a couple more problems...
Here they are:
Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 196 Hz. One of the strings is retuned by adjusting its tension. When this is done, 1.5 beats per second are heard when both strings are plucked simultaneously. I found the lower and upper limits on the possible fundamental frequencies of the re-tuned strings to be 194.5 and 197.5 Hz, respectively. I'm struggling with this:
By what fractional amount was the string tension changed if it was increased?
By what fractional amount was the string tension changed if it was decreased?
(delta T)/T

Could someone help me with this? Thanks!

Frequency depends on the square root of the tension. Since you have the frequencies and the original tension, I think that's enough to solve it.

 

[M98Ranger]

It sounds like you have encountered some interesting problems with frequency and tension in regards to violin strings. As for your questions, determining the fractional amount of change in tension can be a bit tricky but here is a possible approach to solving it. First, we can use the formula for beats per second, f_beat = f_2 - f_1, where f_2 is the frequency of the retuned string and f_1 is the original frequency of 196 Hz. We know that 1.5 beats per second were heard, so we can plug that in and solve for f_2. This gives us f_2 = 197.5 Hz.

Now, we can use the formula for fundamental frequency of a string, f = (1/2L)*sqrt(T/m), where L is the length of the string, T is the tension, and m is the mass per unit length. Since the strings are identical, we can assume that m is the same for both strings. We also know that L is constant and does not change. So, we can set up the following equation:

f_1 = f_2 = (1/2L)*sqrt(T_1/m) = (1/2L)*sqrt(T_2/m)

Solving for T_2, we get T_2 = (f_2/f_1)^2 * T_1. Plugging in our values, we get T_2 = (197.5/196)^2 * T_1 = 1.0001276 * T_1. This means that the tension was increased by approximately 0.01276% (0.0001276) in order to produce a frequency of 197.5 Hz.

Similarly, if the tension was decreased, we can use the same formula and solve for T_2. We would get T_2 = (194.5/196)^2 * T_1 = 0.99795 * T_1. This means that the tension was decreased by approximately 0.00205% (0.0000205) in order to produce a frequency of 194.5 Hz.

I hope this helps and good luck with your further exploration of frequency and tension in violin strings!