Work done by internal forces on a system of particles

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In analyzing the work done by internal forces in a two-particle system, calculations yield different results based on the chosen frame of reference, such as the center of mass or one particle's frame. This discrepancy is normal, as work is frame-dependent, but energy conservation holds within each frame. The total energy of the system remains constant if no external work is applied, regardless of the frame of reference. To compare total energy across different frames, one must ensure that calculations account for the same system dynamics. Understanding these principles clarifies the relationship between work, energy, and reference frames in physics.
Nikhil Rajagopalan
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Dear Experts,
Trying to analyze the work done by internal forces in a very simple two particle system which is attracted to each other with a constant force, i performed calculations based on two frame of references, 1.The center or mass frame of reference and 2. Frame of reference of one of the particles say A.
Computing the work done by internal forces on a system of particles, under two different frame of references, i obtain two different answers.I have uploaded the image of the derivation.

The answers for the change in kinetic energy will be hence different in both frame of references. Is that normal?

What are the cases where the work done by internal forces becomes 1.zero, 2. positive, 3. negative ?
 

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Nikhil Rajagopalan said:
Computing the work done by internal forces on a system of particles, under two different frame of references, i obtain two different answers.
That is correct. Work is frame variant.

Nikhil Rajagopalan said:
The answers for the change in kinetic energy will be hence different in both frame of references. Is that normal
Yes. The important thing is that energy is conserved within each single frame. It doesn't matter that it is different from frame to frame.
 
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Thank you Dale, i understand that the work depends on the frame of reference.
But is it true that the total energy of the system of particles must remain constant if there is no external work done on it, irrespective of the frame of reference. Is there any way i can compare the total energy of the system in different frame of references and show that they are all equal. Considering the problem i worked on in the image shared ?
 

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