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**1. The problem statement, all variables and given/known data**

A mass M is connected between 2 stretched springs, as shown. The equilibrium length of each spring is L, they are stretched to twice this length when m is in equilibrium. Consider the oscillations in the + and - x direction as shown in the figure:

http://img22.imageshack.us/img22/4946/moochka.jpg [Broken]

http://g.imageshack.us/img22/moochka.jpg/1/ [Broken]

a) using the energy method, find the angular frequency of oscillation of the mass (you will need to use binomila approximation twice )

b) find the angular frequency using the equation of motion method (you have to make an approximation in which you only keep the leading term of x)

**2. Relevant equations**

Restoring force of a spring is F = -kx

**3. The attempt at a solution**

I (think) this method i'm using is the equations of motion one:

each spring will exert a force of (stretched length - L) x k

From the triangles in the picture

Strentched length = root (x^2 + 4L^2) (the diagonal the spring would make with the mass at a poiint x)

Force due to each spring = k * (root (x^2 + 4L^2) - L)

Isolating only the component in the negative x direction:

Fx = - k * (root (x^2 + 4L^2) - L) * x / (root (x^2 + 4L^2)

Doubling the force due to there being 2 springs

Fx = - 2k * (root (x^2 + 4L^2) - L) * x / (root (x^2 + 4L^2)

a = F/m = - 2k/m * (root (x^2 + 4L^2) - L) * x / (root (x^2 + 4L^2)

since it will oscillate w/ SHM, the acceleration is also d^2x / dt^2 = -w^2x

setting the 2 expressions equal to each other and cancellin the x's and -1s we get

2k/m * (root (x^2 + 4L^2) - L) * 1 / (root (x^2 + 4L^2) = w^2

If i solve for w i would get it as a functino of x at this point...is this how it's supposed to be?

I get somethign like w^2 = 2k/m * (1 - L/root (x^2 + 4L^2))

I don't see where i'm supposed to make an approximation with keepin the leadin term of x...or how to proceed from this point...and I've no idea how to even start deriving w by the "energy method"?

Help!

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