Work done by internal and external forces on a system of particles. Why can we use $dW=\vec{F}\cdot d\vec{s}$ for some systems of particles?

Question

I know given a system of particles that the net external force on all the particles in the system is equal to the total mass*the acceleration of the centre of mass Which can be derived by applying Newton’s third law to the system.

Is the work done by the internal forces always zero? If not how can I prove it? Under what conditions will it be zero (For example the work done by internal forces in a rigid body are always zero)?

Why can I use dW=F.ds to calculate work done in pushing a column of fluid in u-tube or calculating work done in compressing a ideal gas or potential energy stored in an elastic solid.even though as of my understanding the work done = change in kinetic energy can be derived for a point mass (can it be derived for the above cases?)

Answer 1

Is the work done by the internal forces always zero? If not how can I prove it?

If the forces are internal to the "system" as defined, and your asking if work is done on the system, then the answer is zero . But it depends on how you define the system.

Consider two ice skaters initially at rest in casual contact with one another while standing on the ice. Assume that there is no friction between their skates and the ice. Skater 1 applies applies a horizontal force $F$ on skater 2. Per Newton's 3rd law skater 2 exerts and equal and opposite force $F$ on skater 1. Is the force $F$ internal or external? It depends on how you define the system.

If we consider the two skaters as constituting a two particle system and ask if the force they apply to one another does work on the system, then the answer is no as the forces they apply to one another would be considered internal forces. While each skater individually accelerates away from the other per Newton's second law, the center of mass of the two skater system remains stationary. There is no work done on the two skater system.

On the other hand if we consider skater 2 alone as the "system", or skater 1 alone as the "system", then the force $F$ they apply to one another is "external" to each, doing work equal to the change in kinetic energy of each.

The "proof" of whether or not work is being done on the "system" as defined, is whether or not there is a change in momentum of the system. .

Hope this helps.