The work done by two objects on each other

[John Constantine]

TL;DR Summary

The work done by two objects on each other.

The work done by A on B as Wab, and the work done by B on A as Wba.

I have always thought that Wab = -Wba generally always holds true. Looking at it from the perspective of energy transfer:

1. "A do 10J of work on B."
2. "A transfer 10J of energy to B."
3. "B received 10J of energy from A."
4. "B do -10J of work on A."

If 1 is true, then 2 through 4 all hold true, making Wab = -Wba valid.

However, consider the case where two people of different masses push each other from rest. In this scenario, it is intuitively clear that Wab and Wba are not the same. The kinetic energy is not conserved in this case.

I asked about this on another physics forum, and the response was that sometimes it is true (e.g., in the case of a perfectly elastic collision) and sometimes it is false (e.g., when two objects of different masses attract each other due to gravity, or when two stationary objects of different masses push each other). In other words, the statement is sometimes correct and sometimes incorrect. What conditions are necessary for this statement to be true, and what conditions make it false?

 

[PeroK]

Newton's third law involves forces and momentum. In general, it's the force (hence total impulse = change in momentum) of A on B that is equal in magnitude and opposite in direction to the force (hence impulse) of B on A.

This is not true for energy. In general, $W_{ab} = - W_{ba}$ is not true.

 

[jbriggs444]

However, consider the case where two people of different masses push each other from rest. In this scenario, it is intuitively clear that Wab and Wba are not the same. The kinetic energy is not conserved in this case.

I asked about this on another physics forum, and the response was that sometimes it is true (e.g., in the case of a perfectly elastic collision) and sometimes it is false (e.g., when two objects of different masses attract each other due to gravity, or when two stationary objects of different masses push each other). In other words, the statement is sometimes correct and sometimes incorrect. What conditions are necessary for this statement to be true, and what conditions make it false?

One first needs to be clear about what sort of work is being considered.

Is it net work (net force times the [parallel] displacement of the center of mass)?

Is it mechanical work (individual force times the [parallel] displacement of the material of the target at the place where a particular force acts.

Based on the scenarios that you have called out, it appears that you are considering a particular interaction force times the [parallel] displacement of the center of mass. This is what I would call the "center of mass" work associated with a particular force.

For a force pair to do no "center of mass" work, the component of the relative velocity between the two centers of mass in the direction of the force pair must be zero.

Two balls springing apart have non-zero relative motion. Non-zero work is done by the repulsive force pair.

Two planets revolving about one another in a circular orbit have zero relative motion (along the force axis). The attractive force pair does zero work.

A hand on the end of a wrench has zero motion relative to the wrench. The force pair between hand and wrench does zero work.

A sanding block and a slab of walnut have non-zero relative motion in the direction of the frictional force between them. The frictional force pair between block and wood absorbs non-zero work.

A sanding block and a slab of walnet have zero relative motion in the direction of the normal force between them. The normal force pair between block and wood does zero work.

A man holding the center of mass of man+hand+sanding block steady as he moves a sanding block over the walnut has zero relative motion in the direction of the frictional force between block and a stationary slab of walnut. The frictional force pair between man+hand+block and wood does zero center of mass work.

A car driving along the highway has non-zero relative motion in the direction of the frictional force between car and road. The force pair between car and road does non-zero work.


Turning our attention to mechanical work...

For a force to do no "mechanical" work, the relative velocities of the parts of the two systems that exert force on one another must have zero relative motion in the direction of the force pair.

The contact patch on a drive wheel on an automobile has [near] zero relative motion relative to the road upon which it rides. The force pair between tire and road does [near] zero mechanical work

 

[A.T.]

I have always thought that Wab = -Wba generally always holds true.

A very basic counter example is kinetic friction, that dissipates some of the mechanical energy at the interface, so Wab + Wba < 0.

You can also have Wab + Wba > 0 when mechanical energy is generated from some other form of energy at the interface.

But note that this is very dependent the level of abstraction, or what you include in the interface between the two objects, and whether you consider "real work" or "center of mass work", as mentioned above by @jbriggs444.

 

[sophiecentaur]

TL;DR Summary: The work done by two objects on each other.

However, consider the case where two people of different masses push each other from rest. In this scenario, it is intuitively clear that Wab and Wba are not the same.

Why would it have to be the same? We have to resist looking for patterns that are not there. A will have a net Energy (relative to some frame) which will consist of its KE plus whatever fuel it has in its tank. As a result of an 'equal and opposite' force it will interact with B , which will also have a net energy (KE + fuel). Where the net energy ends up will depend on the details but there is no reason to assume how much energy A and B gain or lose will be equal.
If I do a standing jump on the ground, I will do (virtually ) no work on the Earth. When I start the jump I will have so much KE due to the leg work which will gradually change to PE (relative the Earth). 'All' the work was done "ON" me (if you really want some anthropomorphic thing to be responsible. But I supplied all the energy and, as I start to fall down, I steal back 'all' the Potential Energy (the Earth remains stationary). The only energy that the Earth gets back will be some strain energy and some heat. This could be significant if I land in something soft. Did I do the work 'on' myself and 'on' the landing spot? It's a meaningless question.