# Potential energy of a spring thought experiment

Just wondering if I am using the concept of potential energy and conservation of energy correctly in the following thought experiment.

Lets say we take a compressed spring into space. If we attach a mass to each end and than let it go after a while the spring will return to its equillibrium position and the masses will have a certain kinetic energy(lets say the springs kinetic energy in this case is negligible).

Does this mean if I let the compressed spring go with no masses attached, after the spring returns to equillibrium its kinetic energy is equal to that of the masses in the previous case?

My other question is: So this would mean if I could build a very light spring and than let it go in space it would be translating or rotating very fast?

Firstly the spring upon relaxation will start vibrating. Though assymetric release of the spring may induce some rotation but certainly not translation ( centre of mass has to remain stationary as no external force is applied ) . But you are right spring with smaller mass will oscillate with higher frequency. Also in case of masses attached, both the mass will get translational motion hence some kinetic energy will be lost from the spring. Rest of the energy will be associated with the spring's vibrational motion.

• The OP did not ask about an actual spring it's more an experiment of thought. – Felix Crazzolara Jul 26 '17 at 11:27

What is actually the important point in your question is the part in brackets:

(lets say the springs kinetic energy in this case is negligible)

By saying this, you assume the masses attached to the spring being much bigger than the mass of the spring and thus (since the velocity of the sprint isn't bigger then the one of the masses either) the kinetic energy of the spring can be vanished. But if you remove your mass, you can not do any order approximations of this kind anymore. Thus for any real spring with mass bigger than $0$, the energy at it's equilibrium point is stored entirely in the kinetic energy of the spring oscillating. There are no strange rotations or translations occurring.

First off, we need to go over your assumptions. Specifically, the assumption that the spring's kinetic energy is negligible and (unstated) so is its mass. The assumption that its mass is negligible only holds when the mass of the two masses is significantly larger than the spring's mass. The assumption that the spring's kinetic energy is negligible only holds when the kinetic energy of the masses is significantly larger than the kinetic energy of the spring, which is effectively that the spring's mass is negligible.

So, discarding those assumptions, let's analyze all sources of energy: the translational kinetic energy of the masses and the vibrational kinetic energy of the spring (translational kinetic energy of the pieces of the spring), and the potential energy in the spring. There is no potential energy stored in the masses.

When the spring is stretched, it gains potential energy. When you let go of the spring, that potential energy is converted into kinetic energy, some in the spring and some in the masses. When the spring is relaxed, the sum of all those kinetic energies will equal the potential energy of the initially stretched spring. If you were to remove the masses, the spring would have more kinetic energy than just the masses, but equal kinetic energy to the potential energy of the stretched spring.