Schrödinger's Cormorant

3.4 Let’s Work

A reminder about definite integrals. Relating work done to force applied, when the force is not necessarily constant.

Work is the transfer of energy to or from an object by the application of a force to that object to make it move. In fact, the work-energy theorem, in classical mechanics, states that the work done on an object equals the change in kinetic energy of the object as a result.

If we limit ourselves to one dimension to keep it simple, you know that if the applied force (F) is constant, and the distance moved (displacement) is s, you can define work (W) as:

Work = force x distance moved (W = FΔx) III-4

Just for a bit of fun, let's verify the work-energy theorem for this case (constant force):

Question 3-6A

However the force may not be constant (this will often be the case: for example a gravitational or electromagnetic force gets bigger as objects get closer together). Let’s see how we can deal with this, to get a more general expression. (If you’re comfortable with the graphical meaning of integration, you can skip lightly through this bit).

Question 3-7

Let’s take this a bit further. Looking back at Question 3-7, if we take the approach of graph C, we can make the Δx interval smaller and break the overall movement down into smaller steps. If we have n slices, each of width Δx, the general statement we can always make is:

If we keep on making the steps smaller and smaller they eventually become infinitesimal and Δx becomes dx. The sum of all the Fdx elements still add up to the total work and we call this summation of infinitesimal terms an integration. So we can write:

Question 3-8

All of this may be slightly interesting but also pointless: integration is clearly not a necessary tool to evaluate the work done in this case. Where it comes into its own is when the force varies in a non-linear way with distance. For example, consider this:

You can see that the curvature of the line means that using the midpoint over the range from 10 – 30 m, or even breaking it up into 10m slices, would give hopeless estimates of the area. However breaking it up into thinner slices, the errors introduced become smaller. By making the slices infinitesimally thin, the errors are completely eliminated: this is how integration works. Thus our equation for the work done (equation III-5) is a generally applicable one. Here’s an example:

Question 3-9

Now we've got that bit of maths up to speed, let's use if for our next step.

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