Where F is the tension in the string and µ is the linear density (mass per length) of the string.

A violin string vibrates at the fundamental frequency of 512 Hz under the tension of 480 N.
What should be the tension in the string to achieve a fundamental frequency of 460 Hz?
A. 523 N
B. 491 N
C. 431 N
D. 387 N
E. 570 N

Is this C?
M= 460/512= 0.8984375 , For magnification
√T=0.8984375
T= 0.807N , By multiplying the magnification
T= 512*0.807= 413.3N

Yes. The answer is C.

Well, it seems like you did some math there, but let me give you my two cents. When it comes to tension in a violin string, it's like dealing with a hot potato. You want just the right amount to achieve the desired sound, but too much and things might start to get a little too intense.

In this case, if the original tension was 480 N and the frequency was 512 Hz, and you want a frequency of 460 Hz, you'll need to adjust the tension accordingly.

Now, I won't make you wait any longer for my punchline. The correct answer is B. Yes, that's right, B for "Better tune that string!" with a tension of 491 N. So go ahead and reach for the B-string, or should I say, the "B-string of brilliance!"

To find the tension in the string to achieve a fundamental frequency of 460 Hz, you can use the equation:

T = (M²) * (µ * ω²)

where T is the tension in the string, M is the desired frequency divided by the original frequency (460/512 = 0.8984375), µ is the linear density of the string, and ω is the angular frequency.

Since only the tension needs to be found, rearrange the equation to solve for T:

T = (M²) * (µ * ω²)

Given that the tension in the original string is 480 N, use this value to find the angular frequency (ω).

480 = (1²) * (µ * ω²)

480 = µ * ω²

We don't have the value for the linear density of the string (µ). Therefore, we cannot determine the tension required to achieve a fundamental frequency of 460 Hz. None of the provided options are correct based on the information given.

To calculate the tension in the string required to achieve a fundamental frequency of 460 Hz, we can use the equation:

f = (1/2L) * sqrt(T/µ)

Where:
- f is the fundamental frequency in Hz
- L is the length of the string
- T is the tension in the string in Newtons
- µ is the linear density of the string (mass per length)

Given that the initial tension is 480 N and the initial fundamental frequency is 512 Hz, we can calculate the initial linear density:

512 = (1/2L) * sqrt(480/µ_initial)

µ_initial = (480 / (512 * (1/2L))^2)

Now, let's calculate the linear density for the desired fundamental frequency of 460 Hz:

460 = (1/2L) * sqrt(T/µ_desired)

Simplifying this equation, we get:

µ_desired = (T / (460 * (1/2L))^2)

To solve for T, we can set the two expressions for µ equal to each other:

(480 / (512 * (1/2L))^2) = (T / (460 * (1/2L))^2)

Simplifying this equation, we get:

T = (480 * (460 / 512)^2)

Calculating this expression, we find that T is approximately 413.3 N.

Therefore, the option closest to 413.3 N is D. 387 N.

So the correct answer is D. 387 N.