# Sound & Music - Mass per unit length

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1. Feb 17, 2017

### Torrie

1. The problem statement, all variables and given/known data
Suppose a harp string is tuned to middle C (C4) and is 0.56m long. If I want the tension to be 193.5 N, what mass per unit length do I need the harp string to be? Calculate your answer in kg.m

2. Relevant equations
Vs = sqrt(T/μ)
V = fλ

3. The attempt at a solution
Having a very hard time. I looked up the frequency of a C4= 261.61 Hz. I calculated Vs as 261.61 x 1.12 = 293. Then I attempted (293 x 193.5)sq, but that number is way too high. I am stuck.

2. Feb 17, 2017

### TomHart

Why did you multiply 293 x 193.5?

3. Feb 17, 2017

### Torrie

I figured if Vs = sqrt (T/μ), then μ= Vs x T sq

4. Feb 17, 2017

### TomHart

If Vs = sqrt(T/μ), then μ = T/Vs2, doesn't it?

5. Feb 17, 2017

### Torrie

Thats what I thought initially! But when I answered 0.436 kg/m the it said it was incorrect.

6. Feb 17, 2017

### TomHart

I don't see how you got 0.436 kg/m. I got a different answer. Can you write out what you did mathematically.

7. Feb 17, 2017

### TomHart

Okay, I see what you did. You divided 193.5 by 293, and then you squared it. It is not (193.5/293)2. It is 193.5/(293)2.
Or another way to do it is divide 193.5 by 293 and then divide that result by 293 again.

8. Feb 17, 2017

### Torrie

μ = (T/Vs)sq

T = 193.5
Vs = 293

193.5/291 = .66
.664948sq = .4356 kg / m

9. Feb 17, 2017

### Torrie

Oh! Okay let me try that

10. Feb 17, 2017

### Torrie

Okay finally! .0022 kg/m is correct. Can you explain to me why it is 193.3/(293) sq instead of (193.3/293)sq?

11. Feb 17, 2017

### TomHart

Algebraically, that's how the equation turns out when you rearrange it to solve for μ.

Vs = sqrt(T/μ)
Square both sides of the equation to give:
Vs2 = T/μ
Multiply both sides of the equation by μ to give:
μVs2 = T
Divide both sides of the equation by Vs2 to give:
μ = T/Vs2

Does that make sense?

12. Feb 17, 2017

### Torrie

That does make sense!! Thank you so much for all of your help. I really appreciate it!