Sound & Music - Mass per unit length

In summary, if you want the tension on a harp string to be 193.5 N, you need to use a mass of .0022 kg/m.
  • #1
Torrie
29
2

Homework Statement


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

Homework Equations


Vs = sqrt(T/μ)
V = fλ

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.
 
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  • #2
Why did you multiply 293 x 193.5?
 
  • #3
I figured if Vs = sqrt (T/μ), then μ= Vs x T sq
 
  • #4
If Vs = sqrt(T/μ), then μ = T/Vs2, doesn't it?
 
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  • #5
Thats what I thought initially! But when I answered 0.436 kg/m the it said it was incorrect.
 
  • #6
Torrie said:
But when I answered 0.436 kg/m the it said it was incorrect.
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
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
μ = (T/Vs)sq

T = 193.5
Vs = 293

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

This answer was not correct
 
  • #9
Oh! Okay let me try that
 
  • #10
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
Torrie said:
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?
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?
 
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  • #12
That does make sense! Thank you so much for all of your help. I really appreciate it!
 
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FAQ: Sound & Music - Mass per unit length

1. What is mass per unit length in relation to sound and music?

Mass per unit length is a physical property that describes the amount of mass in a given length of a material. In the context of sound and music, it refers to the mass of a string or other vibrating object per unit length.

2. How does mass per unit length affect the pitch of a sound?

The mass per unit length of a string or other vibrating object directly affects the pitch of the sound it produces. A higher mass per unit length will result in a lower pitch, while a lower mass per unit length will result in a higher pitch. This is because the frequency of a sound is inversely proportional to the mass per unit length of the vibrating object.

3. Can the mass per unit length of an object be changed to alter its sound?

Yes, the mass per unit length of an object can be changed to alter its sound. This can be achieved by either changing the material of the object or by altering the dimensions of the object. For example, shortening a guitar string will increase its mass per unit length, resulting in a lower pitch.

4. How does mass per unit length affect the volume of a sound?

The mass per unit length of a vibrating object does not directly affect the volume of a sound. However, it can indirectly impact the volume by altering the pitch of the sound, which can be perceived as louder or softer to the human ear.

5. Is the concept of mass per unit length only applicable to stringed instruments?

No, the concept of mass per unit length applies to all vibrating objects, not just stringed instruments. It can also be observed in instruments such as drums, where the thickness and material of the drumhead affect the pitch and tone of the sound produced.

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