Simple Harmonic Motion with 2 Springs attached to a Dynamics Cart

[DGalt]

I'm trying to write a lab report but I cannot figure out how to derive the formulas for my Theories and Derivations section.
Basically, the set-up was this:
Two springs were attached to either ends of a dynamics cart. Each spring was then attached to a bumper at either end of the dynamics track (basically just something to attach the springs to). The cart was then pulled to one end of the track (compressing one spring and stretching the other) and then released. The time it took the cart to complete five periods was recorded, and we needed to calculate the spring constant for the system from this.

Now, I know how to do the math from the actual lab data. That's not the problem. The spring constants should be additive (they kinda were for our experiment, but we had some pretty bad percent error). Using the equation T = 2(Pi)Sqrt(m/k) we solved for the k of the system.

Now, my problem is proving theoretically that the spring constants of each spring should add to produce the spring constant for the whole system.

The reasoning I've gone through so far is this:
If there were no outside forces, the system would oscillate forever with a constant x for each respective spring. However, since the spring constants for the two springs were different, these x values will be different for each spring. I know that while the acceleration is increasing for one spring, it is decreasing for another. Overall, though, the force should be constant (I think) because the only force that's being applied to the system is that of the moving dynamics cart.

The problem that I'm having is proving that the restoring force is F= (k1+k2)x. In my head I know it is, but my TA is really picky when it comes to our Theories and Derivations and I can't seem to actually figure out what equations will lead me to this final relationship.

I hope I've been clear in what I need help with. Also, if this post belongs in another part of the forum let me know and I'll move it. It just didn't seem to belong in the actual homework section since, well, it's not really a homework problem.

Thanks in advance

 

[Doc Al]

The problem that I'm having is proving that the restoring force is F= (k1+k2)x. In my head I know it is, but my TA is really picky when it comes to our Theories and Derivations and I can't seem to actually figure out what equations will lead me to this final relationship.

The net force on the cart is the sum of the two spring forces. At the equilibrium position, the net force is zero. If you move the cart a distance of x to the right, the left spring pulls with an additional force of -k1x (minus just means to the left) and the right spring pushes with an additional force of -k2x. The total restoring force is thus F = -(k1 + k2)x.

Make sense?

 

[DGalt]

Thanks, that makes sense. I just had to go through the derivation for x = Acos(ωt + φ), which ended up after a bunch of work showing (k1 + k2) / m = ω2 , which is what he wanted