ENERGY AND WORK

Newton published the Principia in 1687. It ran the solar system, the tides, the trajectory of a cannonball. It did not contain the word energy. The word did not enter physics until 1807, when Thomas Young finally gave a name to the quantity everyone had been arguing about for a hundred and twenty years.

A century and a quarter is a long time to be missing a word. And that lag is the whole story. Newton had forces and momenta, but he did not have the thing that gets transferred when a falling brick speeds up, or a coiled spring uncoils, or a steam piston lifts a weight. Leibniz thought he had it — he called it vis viva, living force, and set it equal to m·v². The Newtonians said no, the right quantity was m·v. They fought for two generations.

The person who settled it was not Newton and not Leibniz. It was Émilie du Châtelet, writing in Cirey in the 1740s while translating the Principia into French. She read Willem 's Gravesande's experiments — brass balls dropped into soft clay from different heights — and noticed something both sides had missed. A ball dropped from twice the height sank four times as deep. Not twice. Four times. The quantity that scaled with the damage was v², not v. Leibniz was right, up to a factor of one half. Newton's camp was wrong.

She wrote it up. Voltaire — her lover, housemate, and the most famous Newtonian in France — disagreed loudly. She published anyway. She was dead five years later, at forty-two. The word energy would take another sixty years to arrive, but its defining equation was already on the page: ½·m·v².

Start concrete. Push a crate across a floor with constant force F over a distance d, in the direction of the push. The product F·d is the work you did on the crate. Push twice as hard, or twice as far, and you do twice as much work. Newtons times metres have their own name: joules.

One subtlety hides in that line. Work only counts to the extent force and displacement point the same way. Hold a suitcase steady at arm's length and your biceps scream, but the bag does not move — zero displacement, zero work. Push sideways on a crate that will not budge — zero displacement, zero work. Physics work is not effort. It is force times distance in the same direction.

For a force at angle θ to the path, only the aligned component counts:

That is why a satellite in a circular orbit is not gaining or losing energy: gravity points inward, motion is tangential, cos 90° = 0. The force does no work. The satellite coasts forever at the same speed because no energy is being transacted across the force–motion interface.

When the force varies along the path — a stretching spring, a rocket ramping up, gravity falling off with altitude — the product becomes an integral. The work is the area under the F(x) curve:

For a Hookean spring F = −k·x, integrating from 0 to x gives W = ½·k·x². For constant gravity over a height h, the work done by gravity on a falling body is m·g·h. Those two answers are the entire scaffolding of what comes next.

Do work on a free body and it accelerates. How much? Start with Newton's second law F = m·(dv/dt), multiply both sides by v = dx/dt, and integrate along the path:

The left-hand side is the work done on the body. The right-hand side is the change in a quantity we call kinetic energy:

This is the work–energy theorem: the net work done on a body equals the change in its kinetic energy. Push a 1 kg puck with 10 N across 5 m of frictionless ice — 50 J of work in, 50 J of kinetic energy out. From rest, that is 10 m/s. No extra bookkeeping needed.

The v² is load-bearing. It is why stopping distance scales quadratically with speed — double the car's velocity and you need four times the braking distance, because four times as much kinetic energy has to be dumped into the brake discs. It is why a bullet and a bicycle can have the same momentum and wildly different lethalities. And it is the factor du Châtelet had to fight Voltaire to publish.

Lift a 1 kg brick one metre straight up. You do roughly 9.8 J of work on it. Let go. It falls. By the time it hits the floor it is moving at √(2·g·h) ≈ 4.4 m/s, and its kinetic energy is ½·m·v² ≈ 9.8 J. Exactly the work you did lifting it.

What was the brick storing while it sat motionless on the shelf? Zero kinetic energy — it was still. Yet on command it could produce 9.8 J of motion. That stored something-that-can-become-kinetic-energy is called potential energy. Near the Earth's surface:

The zero of h is arbitrary — sea level, the floor, the centre of the Earth, whichever is convenient. Only changes in PE are physical. What matters is Δ(PE) = m·g·Δh.

Springs store PE too. Compress or stretch a Hookean spring by x from its rest length and it holds:

Release and the stored energy converts back into motion. A diving board, a bow, a pole vaulter's pole, a catapult, a slingshot — all potential-energy reservoirs. You charge them by doing work against a restoring force. They release it on cue.

A force admits a potential energy function exactly when the work it does is path-independent — when the energy is fully recoverable by reversing the motion. Gravity qualifies: lift a brick, lower it, you are back where you started. Springs qualify. Electric forces qualify. Such forces are called conservative. Friction does not qualify: push a block forward through grit, push it back, and the grit is warmer both trips — the energy does not come back. Friction is dissipative.

Tie the two halves together. Drop the brick from rest at height h. At the top, KE = 0 and PE = m·g·h. At the bottom, just before impact, PE = 0 and KE = ½·m·v². The work–energy theorem says ½·m·v² − 0 = W_gravity = m·g·h, so v = √(2gh). That is Galileo's falling-body law, re-derived from energetics in one line.

At every height y in between, the same sum holds:

The quantity on the left is the mechanical energy. As the brick falls, KE rises and PE falls in lockstep, but their sum is pinned. This is conservation of mechanical energy, and it holds whenever only conservative forces act. It comes from a general result that is almost a definition: F = −dU/dx. If a force is the spatial derivative of some function U, then the work it does between two points is U(a) − U(b), path-independent, fully recoverable. Mechanical energy KE + U is then a constant of motion, because the two terms differ only in sign.

Drop a ball from the rim of a bowl. Friction at zero: KE and PE trade back and forth forever — the total bar never shrinks. Turn friction up and the red heat segment grows. Energy leaks out of the mechanical ledger into thermal motion of the surface and the ball. The mechanical total (blue + gold) drops, but — and this is the whole point of the next section — the full total is still exactly conserved.

Friction, drag, viscosity, plastic deformation, inelastic collisions — all remove energy from the mechanical accounting. A hockey puck slides across a rink and stops. ½·m·v² went from some value to zero. Where did it go?

It went into the thermal jiggling of about 10²⁵ atoms — in the ice, in the puck, scattered just above the contact patch. No single atom's motion is visible, but summed over all of them, the energy balance holds exactly. A sensitive thermocouple can measure the ice getting microscopically warmer under a decelerating puck. The energy is not gone. It has changed address.

James Joule spent the 1840s pinning down this equivalence. In his most famous experiment, he dropped a weight from a known height, used a string and pulleys to route the weight's fall into the rotation of a paddle inside an insulated tank of water, and measured how much the water warmed up. The numbers matched: every joule of work the weight could have done ended up as exactly one joule's worth of heat in the water. Mechanical energy and thermal energy are the same stuff.

In 1847, Hermann von Helmholtz, twenty-six years old and still in the Prussian army, pulled every strand together in a pamphlet called Über die Erhaltung der Kraft — On the Conservation of Force. Mechanical, thermal, chemical, electrical, gravitational — all the same conserved quantity, all interconvertible. That essay is where the modern law of conservation of energy begins. Not in 1687, not in Newton, not in Leibniz — but in a Berlin pamphlet written by a young army surgeon.

Work is a total. Power is a rate. A 100 kg lift carrying a person 10 m up does about 9.8 kJ of work whether it takes 5 seconds or an hour. The power — joules per second — is what differs. 9.8 kJ in 5 s is roughly 2 kW. The same job drawn out over an hour is 2.7 W: a nightlight.

This is why engines have power ratings. A 100 kW car engine can pour 100 kJ of work into the car every second — lifting it up a hill, accelerating it, fighting drag. Losses exceed power, the car slows. Losses less than power, it accelerates. At terminal speed on a flat road, drag exactly eats the engine's output and net power on the car is zero.

The unit of power, the watt, is named for James Watt, the Scottish engineer who in the 1760s turned the steam engine from a pumping curiosity into the prime mover of the industrial revolution. Watt — selling engines to customers who still thought in horses — invented the unit of horsepower to benchmark his machines. One horsepower works out to about 746 W: a plausible figure for a strong horse lifting a load steadily.

Energy is conserved because the laws of physics do not change with time. Every experiment you can do today, you could have done yesterday, with the same result. That time-symmetry is the deep reason the energy ledger balances. Its proof waits at the other end of this module, in FIG.10 — Noether's theorem.

But energy is not the only thing that balances. There is another conserved quantity, a vector one, that works even when energy does not — in plastic collisions, in explosions, in rockets shedding mass mid-flight. Newton wrote his second law not as F = ma but as F = dp/dt, where p = m·v is called the momentum. That quantity is conserved because the laws of physics do not change with position. The next topic follows it.