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Homework Help: Would mass oscilate ?

  1. Feb 24, 2006 #1
    Hello,
    In my question,I have two masses ( M ) ,one fixed at +y and the other at -y axis and both have a distance of L from the origine. The third mass (m) is located on the +x axis at the distance of X.

    I know that the gravitational forces are acting on the (m) by both masses (M), The net force is on the x axis toward (-x) and magnitude of 2Fgrav and this force will accelerate the (m) toward equilibrium (Origine) and once it is there the Fgrav =0 but because it has a velocity it will continue until its velocity=0 , So this Fgrav force on the X axis is Restoring force .Therefore,the mass will oscilate. My question is since my equation is
    (x:+2GMmx/(L^2+X^2)^3/2=0) , I cant say this is Simple harmonic oscilation because my equation of motion doesnt just consist of x but x/(...+x^2)^3/2,
    So under what condition for x ,it is possible to say that the motion of the mass (m) can be approximated as Simple harmonic motion ?

    Any idea?

    thanks
     
  2. jcsd
  3. Feb 24, 2006 #2

    qtp

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    if you know that the mass will oscillate back and forth on the x axis then you don't need the y components and you can use trig to isolate the x component of the force. this will give you an equation in x
     
  4. Feb 24, 2006 #3
    Do you mean that since my equation of motion x (2dot)+2GMm(x/(L^2+x^2)^3/2=0, ( for SHM, the equation of motion x(2dot)+W^2x=0 ) I can just say that (L^2+x^2)^3/2 = 1 so x=(1-L^2)^1/2 )

    Is this right ?

    thanks
     
  5. Feb 24, 2006 #4

    Gokul43201

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    Staff Emeritus
    Science Advisor
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    As with most other problems where the approximation to a harmonic oscillator is made, the relevant regime is one of small oscillations, ie: x << L (so that, to first order in L2 + x2 ~ L2 )

    Typically, you write the taylor expansion, and will see that the first term after the linear term is of order 3 in x/L. You can throw away this and smaller terms.
     
    Last edited: Feb 24, 2006
  6. Feb 24, 2006 #5
    thank you very much,I understand now
     
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