The Mechanics and Intuition of Potential Energy

This is the equation we are heading for. Once you have it, a large class of mechanics problems stops needing an integration in time and becomes ordinary algebra. Kinetic energy on the left, potential energy beside it, the work of everything else in the middle, and the same two quantities again at the finish. Not one of those symbols is a new law of physics. They all come out of Newton's second law, and in this chapter we pull them out of it. So, a particle, and the curve it happens to be travelling along. Nothing about that route is special: it bends where it likes, and the particle sits somewhere on it. Several forces act on it, and they change from instant to instant. Right now one of them pushes it forward, another drags it back, and the green arrow is its velocity, which always lies along the curve. Add the forces up and split the resultant into two pieces. The piece across the curve bends the trajectory and does nothing to the speed. Only the piece along it, the tangential one, can make the particle go faster or slower. Let it move, and notice how little stays put. Both forces swing round, the tangential piece changes size, and the speed changes with it. Attacking this by integrating the acceleration directly would be grim. So here is the one piece of kinematics we need. Tangential acceleration is the rate of change of speed with time, and by the chain rule that is also v times the rate of change of speed with distance along the path. Newton's second law along the tangent then reads: the sum of the tangential force components equals m times that acceleration, which is now m v, times d v by d s. Separate the differentials, and the tangential force times a little piece of path sits on one side, m v d v on the other. Now integrate both sides over the whole journey, from the starting point to the finishing point. The right-hand side is an ordinary integral in v alone, so it simply evaluates: one half m v final squared, minus one half m v initial squared. And there is our old friend, arriving unannounced: one half m v squared. Nobody defined it for us. It fell out of an integral, and that is the honest origin of kinetic energy. Give that combination a name: T, the kinetic energy. Our result then says something about the integral on the left. Whatever that integral is, it equals the change in T. The integral deserves a name too. As written it demands the tangential component at every single point, which is a nuisance. But a dot product does that job by itself: dotting the force with the small displacement keeps the part along the motion and throws the rest away. So we define the work done by a force between two points as the line integral of force dotted with displacement, and the theorem reads: the work done on a particle equals the change in its kinetic energy. That is the whole of it. If the force is constant and the path is straight, the integral collapses to force times distance, which is the version most of us met at school. That version is the special case. This one is the rule. Which leaves one uncomfortable question. If all the work done on a body goes into kinetic energy and nowhere else, then where does potential energy live? Lift a book onto a shelf and you have certainly given it something. That is the next chapter.

Here is the question the last chapter left us with. All the work done on a body shows up as kinetic energy, and yet everybody talks about energy stored in a raised weight or a squeezed spring. Reconciling those two statements takes no new physics at all. It takes calculus. Some work integrals are far easier than others. Take the simplest force there is: weight, pointing steadily down, the same everywhere. Here it is, and here are two points to travel between. Carry the particle from the first to the second along this high road. Gravity does some amount of work on the way, and computing it from the definition would mean grinding through a line integral. Now the same trip along a completely different route, dipping low. The length is different, the direction of travel is different at every moment, and yet the work done by gravity is exactly the same number. Because the only thing gravity ever cared about was the drop in height. Not the route. Just that one vertical distance. Here is the general statement. A force is called conservative when it can be written as minus the gradient of some scalar function V. Weight can be. Friction cannot. And for such a force, the fundamental theorem for line integrals does the integral for you. The work from i to f is V at the start, minus V at the end. No integration, no route, just two evaluations. So the work done by a conservative force is minus the change in V, and that function V is what we call potential energy. The minus sign is doing real work: when the force does positive work, the potential comes down. In particle dynamics there are exactly two conservative forces to worry about: weight, and the linear spring. Let us build both potentials from that one definition. Weight first. Here is the floor, and here is a block sitting a height y above it. We agree to call the floor the level where the potential is zero. The only force doing work is the block's own weight. Try the function V g equals m g y, and check it against the definition. Its gradient has only one surviving component, and minus that component is minus m g: precisely the weight vector, pointing down. So weight is conservative, and this is its potential. Now the work done by weight as the block falls. Potential at the start minus potential at the end is m g y initial minus m g y final, which is m g times the height given up. So m g h was never a definition. It is the value of a line integral we no longer have to do, and its shape is fixed entirely by the fact that weight is constant. The other conservative force is the linear spring. Rather than draw the spring, draw its potential: one half k x squared, a parabola, with x measured from the natural length. Sit the block at some compression, here. The force is minus the slope of this curve, and the slope of a parabola grows in proportion to x, so the force is minus k x, pointing back toward the middle. Watch the slope as the compression changes. Steeper further out, gentler closer in, and it would vanish altogether at the natural length. That is exactly what Hooke's law says about a spring. Now the work the spring does as the block is released from a compression d and returns to the natural length. Potential at the start, minus potential at the end: one half k d squared, all of it delivered to the block. That is the whole of potential energy. It is a device for computing the work of the forces whose work does not depend on the route. Friction is not one of them, which is why friction is about to get a term of its own.

Here is the picture nearly every energy problem reduces to. A rough slope, a block on it, weight pulling down, friction rubbing backwards, and a speed that rises as the height falls. Let it slide. Nothing exotic is happening: height is being traded for speed, and friction is quietly taking a cut of the total. Two facts, one from each of the last two chapters. First, the total work done on the body equals the change in its kinetic energy. Second, that total work splits in two. The conservative forces contribute V initial minus V final, and everything else contributes a term we write U prime: friction, a cable, a hand pushing. Both lines describe the same total work, so set them equal to one another. And then move the potentials to the sides they belong on. Initial kinetic, plus initial potential, plus the work of everything else, equals final kinetic plus final potential. That is the equation from the first minute of this lecture, and now every symbol in it has been earned. Read it on the slope: V is the height term, T is the speed term, and U prime is the friction, always negative, because friction always opposes the motion. Nothing here assumed the friction was small, or the path straight, or the forces constant. This is the general form. And if nothing but weight and springs do any work, U prime is zero and T plus V is the same at both ends, which is conservation of mechanical energy as a special case. Two more definitions finish the vocabulary. Power is the rate at which work is being done: how much per second, rather than how much in total. For a machine that is usually what you care about. An engine that can do a great deal of work per second is a powerful engine, and force dotted with velocity is often the quickest way to get at it. Efficiency is the other one. You pour power into a machine, some of it comes out as useful work, and the rest becomes heat, noise and wear. Efficiency is the ratio of the two, and it is always less than one. The same ratio can be taken over a whole job rather than instant by instant, with energies instead of powers. Fill a tank with fuel, and ask how much of it came back as useful work. That is the entire theory: one master equation, and two bookkeeping definitions. Now watch what it does to a problem that is genuinely painful with Newton's laws.

This problem appeared in the chapter on Newton's laws, where it took two pages. Two blocks on a rough floor, one attached to a spring and one merely leaning against it. Push the pair back a distance d, let go, and ask how big d has to be for the second block to leave the first behind. Here is the arrangement. Wall on the left, a floor with kinetic friction coefficient mu k, the spring, block A glued to its end, and block B just touching A. The dashed line marks the natural length, where x is zero. Push the pair back until the spring is compressed by d, and hold them there. Nothing is moving, so the kinetic energy is nothing, and the spring is holding one half k d squared. Now let go. The spring pushes both blocks forward, friction rubs backwards on both, and the pair speeds up until the spring reaches its natural length. Right there the spring stops pushing. And that is the moment of separation, for a reason worth pausing on. Past this line the spring is stretched, so it starts pulling A back. It can pull A, which is attached to it. It cannot pull B, which is only in contact. So from that instant A slows, turns round and comes back, while B carries on with whatever speed it already had. They part company. Which collapses the whole question to a single number: the shared speed at the natural length. If that speed is greater than zero, B separates and keeps going. If it is zero, nothing separates at all. And that is exactly the sort of question energy answers well: an initial state, a final state, and no interest whatsoever in what happened between them. With Newton's laws you would have to solve a differential equation for a force that changes with position, and only then evaluate it. Put the blocks back where they started, compressed by d and stationary, and write down the master equation. Then take its five terms one at a time. Initial kinetic energy first. The blocks are being held at rest, so T initial is zero. Initial potential next. Gravity contributes nothing, because nothing changes height on a level floor. The spring contributes one half k d squared, the result we derived a moment ago. Final potential. At the natural length the spring is undeformed, so V final is zero as well. And final kinetic energy. At that instant the blocks are still moving together with one shared speed, so it is one half the total mass, times that speed squared. Which leaves friction, and friction is not conservative: there is no potential for it, so we compute its work directly. The floor has to hold up both blocks, so the normal force is the total weight, m A plus m B, times g. Kinetic friction is mu k times that normal force, and it points leftwards the whole way, because the blocks are travelling rightwards. Force against motion means negative work. It is minus the friction force times the distance travelled, which is minus mu k, times the total mass, times g d. Notice that this is the one place where energy saves us nothing. For a non-conservative force you always do the work integral yourself. It is simply that this particular integral is trivial: a constant force over a straight run. Now put the five pieces into the master equation, and double both sides to clear the halves. Zero, plus k d squared, minus twice mu k times the total weight times d, equals the total mass times the shared speed squared. Solve for the speed squared. Divide both sides by the total mass, and there it is: k d squared, minus twice mu k times total mass times g d, all over the total mass. B separates only if that speed is genuinely positive. The denominator is positive already, so the whole condition sits in the numerator: k d squared has to beat twice mu k, total mass, g, d. Every compression is positive, so divide one factor of d out of both sides. The answer: d must be bigger than twice mu k, total mass, g, over k. Compare that with the route through Newton's laws. There you write F equals m a for the pair, with a spring force that changes with position, integrate to get speed against position, and only then set the speed to zero. Same answer, several times the work. And that is the shape of every energy problem. Pick the two states, write T plus V at each of them, add the work of anything non-conservative, and solve. Almost all of the mechanics is in choosing the two states well.