Simple Harmonic Motion - oscillation b/w 2 stretched springs

theneedtoknow

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!

Last edited by a moderator:

Doc Al

Mentor
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)
Good. At this point I would attempt to simplify this expression and then make use of the binomial approximation. Hint: x^2 + 4L^2 = 4L^2(1 + x^2/4L^2).

You want to end up with an expression that you can compare with the standard SHM equation: F = -kx, which you presumably know the solution of.

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