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The plot of a linear relation given an equation

  1. Feb 1, 2015 #1
    1. The problem statement, all variables and given/known data
    [tex]\omega (q)= \sqrt{( \frac{4f}{m})} sin\frac{qa}{2}[/tex]

    7KOmDXz.jpg

    HGXbz3y.jpg

    2. Relevant equations
    N/A

    3. The attempt at a solution
    I dont understand the linear line given on the graph. For low q (or as q tends to zero) it says the relationship is linear. But as q tends to zero for the given equation I dont see how the equation goes to:

    ##\omega (q)= qa \sqrt{\frac{f}{m}}##

    please could someone explain this, it's not a homework question, just going through the notes we have been given.

    thanks in advance for any help
     
    Last edited by a moderator: Feb 1, 2015
  2. jcsd
  3. Feb 1, 2015 #2

    ehild

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    When the argument of the function sin(x) is small, x<<1, sin(x)≈x. So ##ω(q)=\sqrt{\frac{4f}{m}} \sin(\frac{qa}{2}) = \sqrt{\frac{4f}{m}} \frac{qa}{2}##.
    Pull out 4 from the square root, ##ω(q)=2\sqrt{\frac{f}{m}} \frac{qa}{2}##, simplify by 2.
     
  4. Feb 1, 2015 #3
    cant believe i couldnt see that! thanks very much for your help!
     
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