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28 March 2011

An unusual formulation for the work-energy theorem

Most textbooks define work and kinetic energy, then show the connection: the net work done on an object is equal to the change in that object’s kinetic energy.

My students have always had difficulty with this principle. I spend a LOT of time explaining how to calculate the net work on an object, either by taking the scalar sum of the work done by each of several forces, or by calculating the work done by the net force. It doesn’t matter – students will continue to set the work done by any old force equal to a change in kinetic energy (or sometimes to just kinetic energy, without the “change” part.)

The most confusing type of problem asks something like, “How much work must you do to raise a 1 kg block 1 m high?” Students don’t know whether the answer should be +10 J, -10 J, or zero. And don’t get me started on “How much work must be done by an external force to move a charge from point P to infinity.”

These problems are tricky because of the implicit assumption of zero kinetic energy at each position. Using the standard textbook definitions, I can get students to understand that since there’s no change in kinetic energy in these cases zero net work was done. But that doesn’t answer the question! This idea of work done by a non-conservative force is difficult in the context of the work-energy theorem.

My answer this year: I didn’t technically teach the work-energy theorem, as described in textbooks, at all.

I defined work properly, including how to find the sign of work done by a force. (If the force has a component parallel to displacement, work done by the force is positive; if the force component is antiparallel to the displacement, work done by the force is negative.) I defined kinetic energy, and gravitational potential energy.

Next I jumped straight into the conservation of energy, but I wrote it in a weird way. Consider an object moving between two positions A and B. We know the total mechanical energy at position A must equal the total mechanical energy at position B, less the work done by any non-conservative force that dissipates mechanical energy to thermal or other energy. Instead of using one side of an equation for position A and one side for position B, I wrote the following:

WNC = (KEBKEA) + (PEBPEA)
How did I define this WNC term? I called that the work done by a non-conservative force. (How did I define a “non-conservative force?” At this stage, I said a non-conservative force was anything aside from the force of gravity. Bear with me.)

In the standard conservation of energy without friction problem, WNC goes to zero. If a block slides with some friction or air resistance, then WNCbecomes a negative value; if we’re talking, say, an airplane with a propeller, then WNC is the (positive) work done by the propeller. The class had little trouble with this formulation, especially as they quickly recognized that most of these terms will usually go to zero.

As we discussed other forms of potential energy – elastic potential energy and electrical potential energy – we merely changed our equation for PE, and we called WNC the work done by any force that isn’t gravity, a spring, or electricity. This definition is entirely correct and consistent as long as we stick to problems involving a single form of potential energy. (At higher levels of physics we’d have to discuss the deeper meaning of a “conservative” force and how the potential energy is defined… but we’re not at a higher level of physics right now.)

The main advantages of this formulation of what I now call the “work energy theorem:”

• I spend essentially no time explaining how to find the NET work on an object. The concept of net work is not particularly important in this formulation. Good – that used to be confusing to my class.

• We only have to learn and use one overriding equation for all energy conservation-type problems. I used to teach Wnet = ΔKE, plus conservation of energy without friction, plus energy conservation with friction. That’s all accounted for in my newly formulated work-energy theorem.

• The problem in green in the third paragraph above no longer causes trouble! Because I’ve explicitly included how work done on an object can change both the object’s kinetic AND potential energy, there’s no confusion. Both KE terms go to zero, one of the PE terms goes to zero, and we end up with WNC = PEB. When we lift the object, we have to do work equal to mgh; when we move the charge from point P to infinity, since potential energy at infinity is zero, we do work equal to qV at point P.

My colleague teaching honors freshman physics tried this formulation, and he loved it. I’m going to do this again next year. Try it! Tell me what you think.

GCJ

1 comment:

1. This is cool, I like the focus on what the NC forces can do.

What I do is point out that every force is conservative from its microscopic origins. Then I say sometimes we're just too lazy to keep track of everything (all particles, their PE, KE, etc) and so we use the concept of a NC force.