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Suppose in a frictionless environment you perform 20 N*m of work northeast, then 40 N*m of work southwest, then 20 N*m of work northeast, and you're back where you started. (Note that this means the object started at rest, accelerated for a while, slowed back down and started going the other way, then slowed back down again.) From here, there are two different interpretations of the sum of that work.

Net work is just looking at the initial state and the final state and observing the changes. This is the vector sum of the work; in this case, 0 N*m. This measures the change in the overall state of the system.

Total work is the path integral of power (that is, the dot product of the force vector and the velocity vector, which is a scalar) over the trajectory of the object. You can think of it as chopping up the force over infinitely many infinitely small segments of the distance and adding up the scalar magnitudes of those pieces. That gives you 80 N*m. This measures the effort expended by whatever it is doing the work.

If you take that same chopped-up description of the forces that were acting on the object and average them all out, then you could call that the average force. If you started with a system in the same initial state, and you applied this average force vector to the object for the same duration of time, then you would end up with a system in the same end state.

This is actually WHY net work is equal to change in kinetic energy. This description represents the simplest possible way to transform the initial state into the final state. If you were to add any other forces to this description, you would have to add balanced opposing forces to push it back, or else you wouldn't end up in the same end state. And since those forces have to be balanced, they can't possibly cause a change in kinetic energy. That means the only change in kinetic energy that actually matters in the end is that average force over that final displacement.

(Footnote: The sum of infinite pieces thing is called an integral. It's the continuous expansion of the idea of a sum of discrete quantities. It's one of the two basic operations of calculus.)
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Mega Man: The Light of Will (Mega Man / Green Lantern crossover: In the lead-up to the events of Mega Man 2, Dr. Wily has discovered emotional light technology. How will his creations change how humankind thinks about artificial intelligence? Sadly abandoned. Sufficient Velocity x-post)
Old Posted 01-24-2017, 08:31 PM Reply With Quote