Length of string on instrument to produce specific frequency

[idkididk]

Homework Statement

You are designing a two-string instrument with metal string 35.0 long, as shown in the figure . Both strings are under the same tension. String has a mass of 8.45 and produces the note middle C (frequency 262 ) in its fundamental mode.

To extend the range of your instrument, you include a fret located just under the strings, but not normally touching them. How far from the upper end should you put this fret so that when you press tightly against it, this string will produce C# (frequency 277 ) in its fundamental?

Homework Equations

f= (1/(2L)) * (F/u)^.5, where f is the frequency, L is length of the string, u is the mass per unit distance, and F is the tension force.

The Attempt at a Solution

So I found that the tension of the string to be 812N. Using that, I solved for x: 277= (1/(2x)) *(812/(.00845/x))^.5

x= .313, which means 31.3 cm. It's asking how far form the upper end, so 35-31.3cm = 3.7cm.

However, this is not correct. Anyone know why?

 

[Doc Al]

So I found that the tension of the string to be 812N.

How did you solve for that? What units are the length and mass given in?

Edit: I see. Length in cm and mass in grams.

 

[Doc Al]

So I found that the tension of the string to be 812N. Using that, I solved for x: 277= (1/(2x)) *(812/(.00845/x))^.5

8.45 is the mass of the full string, so don't divide by the new (smaller) length to find μ. (μ is a constant for the string material, regardless of length.)

 

[idkididk]

^I just tried that, 277= (1/(2x)) *(812/(.00845/.35))^.5, and got x=33cm, so the answer would be 2, but that's not it either

EDIT: Ok nevermind the answer is 1.9 lol. Thanks!

 

[asteriatic]

I would first like to acknowledge that your attempt at solving the problem is a good start. However, there may be a few factors that could have led to your incorrect answer.

Firstly, it is important to consider the effect of the fret on the length of the string. When you press the string against the fret, it effectively shortens the length of the string. This means that the length of the string to be used in the equation should be the distance from the upper end to the fret, rather than the entire length of the string (35.0cm).

Secondly, it is important to ensure that all units are consistent in your calculations. In your attempt, you have used the mass per unit distance (u) in units of kg/m, while the tension force (F) is in units of N. This could lead to an incorrect solution. It is recommended to convert all units to SI units (m, kg, N) before solving the equation.

Lastly, it is important to consider any other factors that may affect the frequency of the string, such as the thickness and material of the string. These could also have an impact on the correct placement of the fret.

In conclusion, to accurately determine the placement of the fret, it is important to consider the effect of the fret on the length of the string, ensure consistent units, and take into account any other factors that may affect the frequency of the string. Further experimentation and testing may also be necessary to fine-tune the placement of the fret for optimal frequency production.