SHM of Mass Oscillating between Two Identical Springs

[SpringPhysics]

1. The problem statement, all variables and given/known
A mass m is connected between two identical springs (along a y-axis) with identical spring constants k. The equilibrium length of each spring is L, but they are stretched to twice this length when m is in equilibrium. By analyzing the force acting on the mass when it is displaced by a small distance x, find the equation of motion of the mass and thus find the angular frequency of oscillation. Ignore gravity.

Hint: You will have to make an approximation, but you only need to keep the term that is linear in x, higher powers of x should be ignored.

Homework Equations

F_sp = -kx = ma

The Attempt at a Solution

When you displace the mass a small x from equilibrium, the y components of the spring forces cancel, leaving twice the spring force in the x direction.

If the angle made from the equilibrium (horizontal) to the stretched state is $$\theta$$ and the displacement is positive x, then

- 2F_sp sin$$\theta$$ = ma_x

I'm really confused about what exactly the spring force is: do I need to consider that the spring is stretched or is that irrelevant? And how does the approximation apply to x (aren't we approximating $$\theta$$?). Can someone lead me in the right direction?

EDIT:
I found the magnitude of the spring force to be k(2L-L) = kL.
I approximated sin$$\theta$$ to be $$\theta$$ = x/2L
So then the net force in the x direction on the mass = -2F_spsin$$\theta$$ = -kx = ma_x
So then the equation of motion is just the differential equation for SHM for a mass on a spring?

Is this correct?

 

[ehild]

Sketch the arrangement, please.

ehild

 

[SpringPhysics]

http://webserv.kmitl.ac.th/~physics/mb/images/stories/publisher/witoon/Problem_oscillation.pdf

Essentially question #53, except with springs.

 

[ehild]

Your solution is all right.

ehild

 

[SpringPhysics]

Great, thanks!