Calculate Final Speeds Two Mass Spring System

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Discussion Overview

The discussion centers on calculating the final speeds of two masses on either side of a compressed spring when released in a frictionless environment. Participants explore concepts related to energy conservation, momentum conservation, and the dynamics of harmonic oscillators, while debating the appropriateness of various models for this scenario.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the situation resembles a perfectly elastic collision, where the potential energy in the spring relates to the masses involved.
  • Another participant introduces the concept of harmonic oscillators and references the governing equations for displacement, indicating that the scenario can be analyzed using Newton's second law or conservation of energy.
  • There is a suggestion to consider the scenario as the reverse of a perfectly elastic collision, likening it to an explosion.
  • A participant notes that while harmonic oscillator math provides frequency of oscillations, it may not directly yield the speeds of the masses, especially since the masses are not attached to the springs.
  • Conservation of total energy is emphasized by multiple participants, stating that the energy from the compressed spring will equal the kinetic energy of the masses upon release.
  • Momentum conservation is also highlighted as a key principle in the analysis.
  • One participant mentions that using conservation equations to find velocities at the moment of detachment from the springs is simpler than solving differential equations.
  • Concerns are raised about the applicability of oscillator math, particularly when the masses are unequal.
  • A later reply suggests disregarding the oscillator approach entirely and focusing on the explosion analogy instead.

Areas of Agreement / Disagreement

Participants generally agree on the conservation of energy and momentum as fundamental principles in this scenario. However, there is disagreement regarding the relevance of harmonic oscillator models and the implications of unequal masses, leading to multiple competing views on how to approach the problem.

Contextual Notes

Participants express uncertainty about the applicability of harmonic oscillator equations in this context, particularly regarding the assumption of fixed walls and the treatment of unequal masses. There are also unresolved mathematical steps related to deriving velocities from energy and momentum conservation.

PCB
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I am trying to calculate the final speeds of two masses on either side of a compressed spring, when the spring is released (in a frictionless environment). The problem has similarities to a perfectly elastic collision, in that the potential energy in the compressed spring would be the result of the v and m of the two masses which compressed the spring. Any suggestions?
 
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A spring with a mass is a harmonic oscillator, which is governed by a second order ODE for the displacement. Your case is simply two harmonic oscillators connected using the same spring.

Here is an example that explains your case.
http://vergil.chemistry.gatech.edu/notes/ho/node2.html
You can get the equations either by applying Newton's second law or conservation of energy.
 
PCB said:
Any suggestions?
What's conserved?

You can think of this as the reverse of a perfectly elastic collision--an explosion.
 
Last edited:
Thanks for the replies. 1. Unless I am wrong, the harmonic oscillator math will just give the frequency of oscillations, but not the speed of the masses (further assume the masses are not attached to the springs, so no oscillations occur). 2. Total energy of the system is conserved, of course. The energy that went into compressing the spring will equal the energy of the moving masses when the spring is released. 3. As you can see from my orginal post, I am thinking of this situation as a PEC, more specifically, the post collision part of the PEC
 
PCB said:
2. Total energy of the system is conserved, of course. The energy that went into compressing the spring will equal the energy of the moving masses when the spring is released.
Good. What else is conserved?
 
Ok, I play the game. Energy and momentum are conserved
 
The harmonic oscillator math will give you x(t) of the masses. Once you have that, you can calculate the velocity by derivation. The governing equations are the same for the fixed and the free mass problem up to the point where you reach the maximum spring displacement.

I also just realized that when you only need the velocities at the moment the masses detach from the springs, using the conservation equations is much easier - no need to solve the ODE's.

EDIT: so yes, just solve the energy and momentum equations as Doc Al suggested
 
PCB said:
Ok, I play the game. Energy and momentum are conserved
That's all you need.
 
Good advice from both of you, thank you. The trouble I am having now is that the oscillator math assumes the spring/mass is acting against a fixed wall. I forgot to specify that my masses are not equal in, er, mass.
 
  • #10
Forget about the oscillator stuff--not relevant. Treat it like an explosion.
 

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