As the title implies, in this lesson, we'll learn through example how even the most abstract and far removed concepts are actually "telling us" a great deal about the physical world. We're going to explore the concepts of work and energy. Specifically, how they are related. I recall in physics class people oftentimes asking: what is energy? Feynman gave the best answer that anyone could possibly give: its a number that we can measure. Energy doesn't come in little blobs or ball bearings, Feynman once said. Energy, like many other concepts in physics—such as angular momentum, the moment of inertia, and so on—is a very abstract, non-concrete notion. This was one of the first big difficulties I had when I first started studying physics. In the beginning we're dealing with concepts that seem concrete and "visualizable" such as position, velocity, and force; but later on in physics, we start to use concepts from which we are unable to form any mental images about.
Fortunately, even as we deal with increasingly abstract concepts in physics, these concepts can still be intuitive. How so you might ask? The answer is that even though the concepts might be very abstract and non-concrete, they become very intuitive once you realize that they can be used in myriad different problems and application to explain how the world works. All of these concepts have "something to say" about how the world works.
Take, for example, the law of conservation of angular momentum. This law, by itself, says nothing more than: if a system is isolated and we measure number, we'll keep measuring that same number. Indeed, this is how Feynman described the law of conservation of energy in his physics lectures. But when we start to look at all the problems the law of conservation of angular momentum can be applied to, we realize that the law is actually telling us something about ho the world should work. For example, suppose an ice skater starts spinning while staying the same spot. Conservation of angular momentum declares that the world should work in such away that \((m\vec{v}×\vec{r})_i=(m\vec{v}×\vec{r})_f\) and you keep measuring the same number. If an ice skater started spinning with her arms out, what does the statement \((m\vec{v}×\vec{r})_i=(m\vec{v}×\vec{r})_f\) predict and tell us would happen if the ice skater brought in her arms towards her chest? Well, her mass obviously didn't change after she brought her arms in, so that must stay the same on both sides of the equation. Her hands were initially pretty far away from her axis of rotation (passing through the center of her body) so that \(r\) must have had been initially pretty big. But when she brought in her hands towards her body, they became much closer to the axis of rotation an \(r\) became much smaller.
The law starts that the number \(mvr\) has to stay the same. If her mass \(m\) didn't chane and the distance \(r\) did change, then the only way for this law to be obeyed—the only way to keep \(mvr\) from changing—js that \(v\) (the tangential speed of her hands) had to change. When \(r\) decreased, \(v\) must have had increased in such a way as to keep the number \(mvr\) the same. Conservation of angular momentum at first seems abstract; but when you apply it to problems, you realize that it actually predicts things about nature that are "visualizable" and concrete. We shall see that the same is also true for concepts such as work, energy, the conservation of energy, and so on. Each of these concepts are "saying something" about how the world works which is visualizable and not at all abstract.
This article is licensed under a CC BY-NC-SA 4.0 license.