Effective Spring Constant

[cowshrptrn]

3 identical masses and four identical springs are set up as shown (m=mass ~=spring |=fixed point)
|~m~m~m~|

the two outside masses are displaced equally inwards. What is the effective spring constant of this system.

attempt at solution:

using FBDs F=-2kx on each individual outside mass
middle mass is stationary at all times, since the two inside springs will cancel each other out, so it acts as if T= 2pi rad(m/2k) (as if it were just |~m~|), but you double the mass for the system (only 2m is moving), but keep the period the same so keff = 4k

i'm not sure how correct this is, or whether the assumptions i made are even correct. This is how a classmate explained it to me, i personally left it as 2k since i couldn't logically justify making it 2m in the period equation since the system didn't follow the standard oscillation I'm used to.

i probably should stic a question here as well, lol

is this correct? It sounds as if I'm missing something, or that using period is the wrong approach to the problem.

 

[kuruman]

The center of mass of the three masses is at the middle mass. If the outside masses are displaced by equal amounts inwards and then released, the center of mass will stay where it is, i.e. the middle mass will be at rest while the other two masses move back and forth mirroring each other. For all they care, the middle mass could be an immovable wall. The frequency of oscillations will be that of mass $m$ connected on either side to immovable walls by two identical springs of constant $k$. What is that frequency?