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String Wave Velocity and Tension

  1. Jun 5, 2014 #1
    1. The problem statement, all variables and given/known data
    The length of a stretched string is 1m. Its mass per length is 2x10-3kg/m. The string resonates at its 4th harmonic at 400Hz.

    (a) Calculate the velocity of the string wave.

    (b) Calculate the string tension.


    2. Relevant equations
    fn = nv / 2L
    v² = F / μ


    3. The attempt at a solution
    I honestly don't know how to go about this.
    (a) v = √F / μ will give me the answer I suppose. μ is the linear density, how do I calculate that? Also for F, which is tension, relates to question (b) and I'm totally unsure how to calculate tension.

    I don't know what nv stands for either, it's pretty hard to Google nv.
    2L is 2 * length of the string, which is 2m.

    I need some serious help please!
     
  2. jcsd
  3. Jun 5, 2014 #2

    CAF123

    User Avatar
    Gold Member

    The string is stretched between two points. This allows for a series of standing waves to be set up and collectively they are called the harmonics of the system, modes of the string that oscillate with a particular frequency. There is a quantisation condition on the allowed wavelengths of these harmonics. That is where the integer n comes in.

    Try to appreciate the physics behind these symbols, it will help you in the long run and to solve problems.
     
  4. Jun 5, 2014 #3
    I'm only an A-level student but here is my analysis on the question:

    a) The 4th harmonic is 400Hz. Harmonics are defined as integer multiples of the fundamental frequency. Therefore the fundamental frequency is 400/4 = 100Hz.
    Think about what that fundamental frequency would look like on the string (i.e. where will the nodes and anti-nodes be) use this information to work out the wavelength of the wave. After that it should be just a case of v = f[itex]\lambda[/itex] to work out the velocity.

    b) Once you know the velocity of the wave all you need to do is rearrange v2= F / [itex]\mu[/itex] to find the tension. They have given you the linear density! It is 2x10-3kg/m.
     
  5. Jun 5, 2014 #4
    Thanks for the responses guys!
    I think I understand how harmonics work. If fundamental frequencies change in 25hz intervals. Then the 3rd harmonic's fundamental frequency would be 75hz etc. But I'm having trouble working out the wavelength λ, to get λ I need to λ = v/f. I'm really confused, if you could give me more insight that would be grand!
     
  6. Jun 5, 2014 #5
    As I mentioned earlier it is easy to work out that the fundamental frequency is 100Hz.

    Now for the fundamental frequency we will have a node at each end of the string and there will be an anti-node in the middle - this is the simplest mode of vibration which is possible. We know that the distance between two nodes is [itex]\lambda[/itex]/2 and we also know that the length of the string is 1m. So combing these two facts we get [itex]\lambda[/itex]/2 = 1 and therefore [itex]\lambda[/itex] = 2m. Now we can use v = f[itex]\lambda[/itex].

    Alternatively we can use the formula you were given (although if you weren't given the formula then I would say that the above method is more intuitive): fn = nv / 2L
    This means that n (the harmonic number) multiplied by the fundamental frequency (f) is equal to n multiplied by velocity divided by two times the string length. Now the value of fn is 400Hz and the value of L is 1m and the value of n is 4 so you could now rearrange this formula to find v.
     
  7. Jun 5, 2014 #6
    Thankyou!
    I had some trouble with the result of (b) through.
    I wrote down the process:
    (a) Calculate the velocity of the string wave.
    fn =
    fundamental frequency * integer n = harmonic resonance
    100 * 4 = 400Hz
    n = 4
    L = 1m

    fn = nv / 2L = 400hz
    400 = 4*v/2*1m = 400hz
    rearrange it to find v.
    v = (fn *2L ) / n = 200hz
    v = (400 * 2) / 4 = 200hz

    v = 200hz
    alternatively
    v = f * λ
    distance between two nodes is λ/2.
    the length of the string is 1m.
    λ/2 = 1
    λ = 2m
    v = 100 * 2
    v = 200hz

    (b) Calculate the string tension.
    v2 = F / μ
    v2 = 40000
    μ = 2x10-3kg/m
    F = tension
    F = v2 * μ
    F =40000 * 2x10-3
    F = 80 // Not sure if correct, wasn't sure on how to go about this
     
  8. Jun 5, 2014 #7
    The numbers you have got seem right to be.

    But be careful v = 200 ms-1 not 200 Hz as you put (Hz applies only to frequency and ms-1 applies to velocity).

    And you should put the units for tension next to your answer for (b). Since tension is a force the units are newtons so your answer to (b) is 80 N :)
     
  9. Jun 5, 2014 #8
    Thankyou so much man! I've actually learned a tonne from you, I feel like I should pay you as my tutor! My greatest thanks!!!
     
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