# Potential Energy defined for internal conservative forces

Will potential energy be always defined whenever internal conservative forces come into play? for example in case of electrostatic interaction between two charged particles potential energy is defined because coulomb's force is conservative. but in case of spring mass system, lets say a spring's one end is attached to the wall and other to a block of mass m. Now if we take spring-mass system, there are two internal conservative forces acting:-

1. forces between the partices of the spring (stress) which correspond to change in potential energy change of spring
2. force that spring and block exert on each other( equal to kx) but this doesn't correspond to change in potential energy of spring? as potential energy is defined as negative of work done by internal conservative forces but here no work is being done by these forces as they exist as action reaction pair and the displacement of the point of action of both these forces is same so they get cancelled.

Am I correct in my reasoning or not? and if I am wrong what am I missing?

I'm not quite sure what your question is, but the action/reaction pair $$kx$$ between the mass and the spring are the consequence of the intermolecular elastic forces between the particles of the spring. In other words, 1. and 2. are not "separate" forces.

Therefore, if $$x$$ is the displacement of the spring from its relaxed position, then the force $$kx$$ corresponds to a change in elastic PE of $$\frac{1}{2}kx^2$$.

If I've misunderstood your question, let me know.

but if we consider a spring+mass system with one end attached to wall and the other with mass m which is elongated by x, then shouldn't the work done on the system cancel out? as equal and opposite internal force acts on both spring and mass but the displacement of point of action is same i.e.,x? then what is causing the rise in potential energy?

It depends on what you mean by "cancel out". While it is true the net work done is zero, the result is elastic potential energy has been stored in the spring/mass system. The following explains how:

Keep in mind that the spring became elongated by $$x$$ because an external (to the spring-mass system) force pulled on the mass, not an internal conservative force. It is that external force doing positive work on the spring/mass system that resulted in the increase in potential energy, as follows:

1. The external force does positive work equal to $$+\frac{1}{2}kx^2$$ because its force is in the same direction as the displacement of the spring. When something does positive work on an object, it transfers energy to the object.

2. The spring does an equal amount of negative work equal to $$-\frac{1}{2}kx^2$$ because its restorative force is always opposite to its displacement. The key point is when something does negative work on an object it takes energy away from the object. In this case, the negative work done by the spring takes the energy supplied by the external force and stores it as elastic potential energy in the mass/spring system.

Hope this helps.

• but if we consider a spring+mass system with one end attached to wall and the other with mass m which is elongated by x, then shouldn't the work done on the system cancel out? as equal and opposite internal force acts on both spring and mass but the displacement of point of action is same i.e.,x? then what is causing the rise in potential energy? plz help I am so confused. Commented Apr 13, 2022 at 14:24
• @GauravKumar I have updated my answer to respond to your comment Commented Apr 13, 2022 at 15:05
• thank u that surely cleared a lot. so lastly i just wanted to ask that is this spring mass system is different from earth-mass system( the mass descends by lets say x) or two charged particles system(the charged particles are free to move due to repulsion) where there is net work done by internal forces(gravitational and electrostatic) as the work done by internal force(action reaction pairs) dont get cancelled out? Commented Apr 14, 2022 at 8:28
• also is it correct to say that the change is potential energy is equal to the work done by internal conservative forces. if not then what is the definition(or expression) for change in potential energy? Commented Apr 14, 2022 at 8:31
• @GauravKumar there are too many follow up questions to cover here in comments. If you wish to discuss them further, we need to do it in Chat. Commented Apr 14, 2022 at 12:25