A string on a violin is tuned to vibrate at a fundamental frequency of 65.0 Hz. If the tension in the string were increased by a factor of three, what would be the new fundamental frequency?
f proportional to sqrt( tension)
so it goes up by 1.73x
To find the new fundamental frequency of the string after increasing the tension by a factor of three, we can use the equation:
f = (1/2L) * sqrt(T/μ)
where:
f is the fundamental frequency,
L is the length of the string,
T is the tension in the string, and
μ is the linear mass density of the string.
In this case, since we only want to find the change in frequency due to the change in tension, we can assume that the length of the string and the linear mass density remain constant. Therefore, we can ignore those factors.
The equation can be simplified to:
f ∝ √T
This means that the frequency is directly proportional to the square root of the tension.
If the tension is increased by a factor of three, we can substitute T' = 3T into the equation:
f' = √(3T)
To find the new fundamental frequency, we need to find the square root of 3T:
f' = √(3) * √T
Now, we substitute the original frequency (f = 65 Hz) into the equation to find the new fundamental frequency:
f' = √(3) * (65 Hz)
f' = 1.732 * (65 Hz)
f' ≈ 112.58 Hz
So, the new fundamental frequency of the string, after increasing the tension by a factor of three, is approximately 112.58 Hz.