Oscillations of a 4-Mass 2-Spring Object. Is This a Difficult Physics Problem?

Rhett Allain
10 min readMay 19, 2024
Image:Rhett Allain. 4 masses and 2 springs

I saw this question posted on AAPT Communities (not sure if you can access that) and it’s a physics trap. It’s question that is presented as something that might be simple, but it might also be super complicated. Here’s the basic idea of the question.

There are four masses connected by the four massless rods as show above. Each mass (m) is connected to the opposite mass with a spring with a constant of (k). Assume the springs do not interfere with each other. What is the period of oscillation if a pair of masses are slightly compressed?

OK, this is both a good and not so good question — and I will get into that in a little bit. However, if this was given to introductory physics students then we are going to have to make some approximations to solve this problem.

  • I’m going to assume small oscillations (that’s obvious from the question).
  • The vertical masses ONLY move in the y-direction and the horizontal masses ONLY move in the x-direction.
  • In order to keep the center of mass at rest at the origin, the motion of opposite masses will be equal and opposite.
  • When the top masses move IN towards the center, the side masses move OUT.



Rhett Allain

Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. WIRED blogger.