Mass Oscillation: Conditions for Simple Harmonic Motion

[esradw]

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 can't say this is Simple harmonic oscillation because my equation of motion doesn't 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

 

[qtp]

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

 

[esradw]

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

 

[Gokul43201]

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 ?

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.

 

[esradw]

thank you very much,I understand now