We call the sum of the weights times the heights gravitational potential energy—the energy which an object has because of its relationship in space, relative to the earth. The formula for gravitational energy, then, so long as we are not too far from the earth (the force weakens as we go higher) is
It is a very beautiful line of reasoning. The only problem is that perhaps it is not true. (After all, nature does not have to go along with our reasoning.) For example, perhaps perpetual motion is, in fact, possible. Some of the assumptions may be wrong, or we may have made a mistake in reasoning, so it is always necessary to check. It turns out experimentally, in fact, to be true.
The general name of energy which has to do with location relative to something else is called potential energy. In this particular case, of course, we call it gravitational potential energy. If it is a question of electrical forces against which we are working, instead of gravitational forces, if we are “lifting” charges away from other charges with a lot of levers, then the energy content is called electrical potential energy. The general principle is that the change in the energy is the force times the distance that the force is pushed, and that this is a change in energy in generaclass="underline"
We will return to many of these other kinds of energy as we continue the course.
The principle of the conservation of energy is very useful for deducing what will happen in a number of circumstances. In high school we learned a lot of laws about pulleys and levers used in different ways. We can now see that these “laws” are all the same thing, and that we did not have to memorize 75 rules to figure it out. A simple example is a smooth inclined plane which is, happily, a three-four-five triangle (Fig. 4–3). We hang a one-pound weight on the inclined plane with a pulley, and on the other side of the pulley, a weight W. We want to know how heavy W must be to balance the one pound on the plane. How can we figure that out? If we say it is just balanced, it is reversible and so can move up and down, and we can consider the following situation. In the initial circumstance, (a), the one-pound weight is at the bottom and weight W is at the top. When W has slipped down in a reversible way, we have a one-pound weight at the top and the weight W the slant distance, (b), or five feet, from the plane in which it was before. We lifted the one-pound weight only three feet and we lowered W pounds by five feet.
Therefore W = ⅗ of a pound. Note that we deduced this from the conservation of energy, and not from force components. Cleverness, however, is relative. It can be deduced in a way which is even more brilliant, discovered by Stevinus and inscribed on his tombstone. Figure 4–4 explains that it has to be ⅗ of a pound, because the chain does not go around. It is evident that the lower part of the chain is balanced by itself, so that the pull of the five weights on one side must balance the pull of three weights on the other, or whatever the ratio of the legs. You see, by looking at this diagram, that W must be ⅗ of a pound. (If you get an epitaph like that on your gravestone, you are doing fine.)
Let us now illustrate the energy principle with a more complicated problem, the screw jack shown in Fig. 4–5. A handle 20 inches long is used to turn the screw, which has 10 threads to the inch. We would like to know how much force would be needed at the handle to lift one ton (2000 pounds). If we want to lift the ton 1 inch, say, then we must turn the handle around ten times. When it goes around once it goes approximately 126 inches. The handle must thus travel 1260 inches, and if we used various pulleys, etc., we would be lifting our one ton with an unknown smaller weight W applied to the end of the handle. So we find out that W is about 1.6 pounds. This is a result of the conservation of energy.
Take now the somewhat more complicated example shown in Fig. 4–6. A rod or bar, 8 feet long, is supported at one end. In the middle of the bar is a weight of 60 pounds, and at a distance of two feet from the support there is a weight of 100 pounds. How hard do we have to lift the end of the bar in order to keep it balanced, disregarding the weight of the bar? Suppose we put a pulley at one end and hang a weight on the pulley. How big would the weight W have to be in order for it to balance? We imagine that the weight falls any arbitrary distance — to make it easy for ourselves suppose it goes down 4 inches — how high would the two load weights rise? The center rises 2 inches, and the point a quarter of the way from the fixed end lifts 1 inch. Therefore, the principle that the sum of the heights times the weights does not change tells us that the weight W times 4 inches down, plus 60 pounds times 2 inches up, plus 100 pounds times 1 inch, has to add up to nothing: