Conservation of Energy

Dennis's mother knew how to turn grams into blocks. For energy we need the rule for each term, and we begin with the energy of a weight raised close to the ground. The argument is Feynman's, modelled on the way Carnot reasoned about steam engines, and it starts from a single postulate: perpetual motion does not exist. On our bench, this means that after a full cycle, with the machines back where they began, no weight may have ended up higher than it was.

Picture a lever that lowers 1 unit of weight by 1 m and, in doing so, raises 3 units to a height \(X\). It is reversible if it works equally well the other way round, taking the unit back up to 1 m while the 3 units come down by the same height \(X\). Every real machine pays some toll to friction, and the reversible one is the ideal case in which that toll is zero.

So take a reversible machine A, which raises the 3 units to \(X_A\), and some machine B, which raises them to \(X_B\). We run B forwards, and the unit goes down 1 m while the stack goes up \(X_B\). Then we run A backwards, the stack goes down \(X_A\) and the unit returns to 1 m. Both machines are back at the start, the unit is again at 1 m, and the stack has finished \(X_B - X_A\) above where it was, with no help from outside.

Were \(X_B\) greater than \(X_A\), repeating the cycle would raise weight without end, and since the postulate forbids this, no machine, however ingenious, outdoes a reversible one. When B is reversible too, we may swap the two roles, and the same reasoning shows that \(X_A\) cannot exceed \(X_B\) either. The height is therefore shared by all reversible machines, and nobody needs to open them up to know it.

We still need the value of that height. Feynman obtains it from an arrangement of shelves in which, on balance, a single unit goes up by \(3X\), and the perpetual motion argument, run in both directions, forces \(3X\) to be exactly 1 m. With the 1 : 3 ratio, then, \(X = 1/3\) m.

In a reversible machine, weight times height stays the same: 1 unit times 1 m before, 3 units times 1/3 m after. The sum of \(W\,h\) over all the weights is the gravitational term in the bookkeeping, the gravitational potential energy, and since the weight is \(W = m\,g\), it is written \(m\,g\,h\). A real machine falls short of the reversible one, and what it loses becomes heat through friction, a term we shall meet later.

In Feynman: §4-2 Gravitational potential energy ↗

Steps 2 to 4 went through the same routine. We pictured a small shift of a system in equilibrium, noted how far each weight goes up or down and required the sum of weight times change in height to come to zero. This is the principle of virtual work, and it serves for mechanisms far more involved than a ramp. In the hinged rod on the bench, the support lets nothing turn, and even so it is an imagined rotation that hands us its force.

We start with the screw jack on the bench, which lifts a 5000 N car. On each full turn of the handle, the hand goes round a circle of radius \(r\) and covers \(2\pi r\), while the screw raises the load by one pitch \(p\), a few millimetres. With \(r = 0.3\) m and \(p = 4\) mm, the hand travels 1.88 m for the car to rise 0.004 m. A zero balance requires \(F \cdot 2\pi r = W\,p\), and the force at the hand comes out at about 10.6 N, the weight of a 1 kg bag of sugar. This is for an ideal screw, since in a real one friction takes a much bigger cut.

The hinged rod is less obvious. It is 4 m long, weighs 30 N, pivots about its left end and carries two weights, of 20 N and 50 N, which you can drag. We want the force \(F\) that a support, at a distance \(x_F\) from the hinge, has to provide to keep it horizontal. In a small rotation \(\delta\theta\), a point at a distance \(x\) from the hinge rises by \(x\,\delta\theta\), and the support rises by \(x_F\,\delta\theta\). A zero balance gives \(F\,x_F\,\delta\theta = \sum_i W_i\,x_i\,\delta\theta\), and \(\delta\theta\) cancels just as the \(s\) of step 3 did.

The rod's own weight counts as if it were all at the centre, at 2 m, because adding up \(W\,x\) over the pieces of a uniform rod gives the total weight times the position of the middle. With the support at 3 m, the sum \(\sum_i W_i\,x_i\) is \(30 \cdot 2 + 20 \cdot 1 + 50 \cdot 2.5\), or 205 N·m, and dividing by the 3 m gives \(F \approx 68.3\) N. Both examples, the jack and the rod, are Feynman's, here with our own numbers.

In Feynman: §4-2 Gravitational potential energy ↗

On the bench, a mass on a 1.2 m string starts from rest 50° from the vertical, 0.43 m above the lowest point of the arc. Let go, it swings down, crosses the bottom at nearly 2.9 m/s and climbs the other side to the dashed line, where it comes to rest again. At the bottom the starting height seems to have dropped out of the sum, yet it returns in full on the way up, so something must have kept it safe in between.

That something is speed, and we can measure it in units of height. A body crossing the bottom at speed \(v\) can climb at most a certain height, and the easiest way to work it out is to think of a stone thrown vertically upwards. From free fall we know that anything falling from a height \(h\) reaches the bottom with \(v^2 = 2gh\), and the same calculation, run backwards, tells us how high the stone gets. Speed is therefore worth as much as a weight \(W\) raised by \(v^2/2g\), and since \(W = m\,g\), \(W\,v^2/2g = \tfrac12 m v^2\). This is the motion term in the bookkeeping, the kinetic energy.

The two bars beside the pendulum follow this from instant to instant, one showing the height lost since the start and the other the current \(v^2/2g\), and they go up and down together. On the graph, the points of \(v^2\) against \(h\) sit on a straight line through the origin with slope \(2g\), about 19.6 m/s².

Readers who can already differentiate will get to the same place without the stone. If \(K = \tfrac12 m\,v^2\), its rate of change is \(dK/dt = m\,\vec v\cdot d\vec v/dt\), and since \(m\,d\vec v/dt\) is the total force, this equals \(\vec F\cdot\vec v\). In the pendulum, the tension in the string is always at right angles to the velocity and adds nothing, while the weight comes in as \(-W\,dy/dt\), where \(y\) is the height of the mass. So \(K\) grows at precisely the rate at which \(W\,y\) shrinks, and the sum of the two terms does not change.

The Galileo's peg button puts a nail directly below the point of suspension. When the string passes the vertical it bends round the peg, and the mass climbs the other side along a shorter, tighter arc than before. Galileo described this experiment in his Discorsi of 1638, and the mass still stops at the dashed line, whatever the depth of the peg, as long as the string does not go slack. The path on the way up has changed and the speed at the bottom has not, and it is the speed that settles the height.

The pendulum, and the idea of measuring the energy of motion by the height it can reach, are Feynman's, while the calculation with derivatives and Galileo's peg are our additions. \(\tfrac12 m v^2\) is also an approximation, and step 7 shows how far it can be trusted.

In Feynman: §4-3 Kinetic energy ↗

On the bench, a 200 N/m spring compressed by 10 cm holds 1 J. Released, it launches a 50 g ball upwards, which rises about 2 m before stopping. As you drag the time slider, the column of bars changes colour, from the elastic share to the kinetic and from the kinetic to the gravitational, but its height does not change. It is the sum kept by Dennis's mother, now with three terms, each with its own conversion rule.

In Braking, a block sliding with 50 J of kinetic energy is brought to a stop by friction, and the 50 J reappear as thermal energy. This term is an old friend under a new name. Friction sets the molecules of the block and the floor moving faster, each in a direction of its own, and thermal energy is the kinetic energy of this disordered motion, which we perceive as two surfaces that have warmed up. The energy is still in the sum, only shared out among an enormous number of particles, and that is why nobody can collect it again to push the block.

In Burning, 50 kJ of chemical energy turn into 45 kJ of heat and 5 kJ of light, the radiant energy. A reaction rearranges electrons and nuclei, and what it gives out comes from the electrical energy between these charges and from the motion of the electrons, which changes along with the arrangement. Elastic energy has the same source, since squeezing the spring brings atoms closer. Even the light from the flame on the bench is produced by charges shaking about in the hot molecules, so the radiant share also ends up on the account of the electrical charges.

In Fission, a uranium-235 nucleus that absorbs a neutron breaks into two fragments, which fly apart with about 200 MeV, mostly as kinetic energy, and all of it kinetic on the bench, to keep things simple. Taken together, the pieces have less mass than the original nucleus plus the neutron, and the difference times \(c^2\) gives those 200 MeV. Nuclear energy comes from a balance between the nuclear attraction, which holds protons and neutrons together, and the electrical repulsion between the protons, and nearly everything fission releases is that repulsion driving the two fragments away from each other. In the bookkeeping it shows up as the mass difference, and by \(E = m\,c^2\) mass itself is a form of energy, one that a body has even when standing still.

Kinetic energy needs revisiting as well. The \(\tfrac12 m v^2\) of step 6 is the first term in the expansion of \((\gamma - 1)\,m\,c^2\), and the two expressions cannot be told apart while \(v\) is small next to \(c\). In Very fast, the two curves are shown side by side, and at v/c = 0.9 the relativistic one is already more than three times the classical one.

This brings us back to the blocks of step 1. Gravitational, kinetic, elastic, thermal, chemical, radiant, electrical, nuclear and mass are terms in one and the same formula, each with its own rule, and all the law promises is that the sum does not change. The list follows Feynman. Reading fission as electrical repulsion is ours, however, since the book treats nuclear energy as a form apart, one that could not be reduced to the others, and the numbers on the benches are ours as well.

In Feynman: §4-4 Other forms of energy ↗