# Effective Spring Constant

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.

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