Conservation of mechanical energy

In brief

Release a pendulum bob from some height and let it swing free. It picks up speed on the way down, slows on the way up, and returns almost exactly to the height it started from, over and over. Roll a ball down one side of a smooth valley shaped however you like, and it climbs the other side to almost the same height it fell from, regardless of the valley's shape. Something is staying constant across all this trading of height for speed, and pinning down exactly what it is turns a problem that looks like it needs the entire trajectory into one you can solve by comparing only two moments.

The full treatment

First look: the pendulum that never overshoots

A pendulum released from rest at some height never swings back higher than that height. If it did, motion would be appearing from nothing, and if it consistently swung back lower and lower, you would want to know where the difference went. In an idealized pendulum, with no air resistance and a frictionless pivot, it returns to almost exactly the release height, cycle after cycle. That repeatability is the clue: whatever quantity is being traded between height and speed is not being created or destroyed, only converted back and forth.

Building the idea: two ways of accounting for the same work

The previous entry defined potential energy as the work a configuration could still deliver, mass times g times height near a surface, or one half k times x squared for a stretched spring. Separately, the work-energy relationship says that the net work done on an object changes its kinetic energy by exactly that amount. Put these together for a falling pendulum bob: as it falls a height h, gravity does work equal to mass times g times h on it, and that entire amount appears as an increase in kinetic energy, since gravity is the only force doing work (the string tension acts perpendicular to the motion and does none). So the kinetic energy gained is precisely equal to the potential energy lost. Nothing about this argument depended on the exact height or the exact speed, only on the fact that the work done by gravity equals the loss in potential energy by definition, and equals the gain in kinetic energy by the work-energy relationship. The two must match.

The formal model: total mechanical energy as an invariant

Define mechanical energy E as the sum of kinetic energy and potential energy at any instant: E equals one half times mass times velocity squared, plus mass times g times height (or the spring equivalent). The argument above says that whenever the only forces doing work are conservative ones, gravity, an ideal spring, an ideal string tension that does no work, E at the start of some interval equals E at the end. Written out for the pendulum: one half m v1 squared plus m g h1 equals one half m v2 squared plus m g h2, for any two points 1 and 2 along the swing. This single equation lets you solve for the speed at the bottom of the swing knowing only the release height, without ever working out the changing angle, changing string tension, or changing acceleration along the arc, all of which do change from moment to moment even though E does not.

Why the path does not matter

The same equation applies to a ball on a curved, frictionless track of any shape, not just a pendulum's circular arc, because the argument never used the shape of the path, only that gravity is conservative, meaning the work it does between two heights is the same regardless of the route taken between them. This is exactly why the framework is powerful: a roller coaster with loops, dips, and hills has a speed at any point determined only by height relative to the release point, not by the twists in between, as long as friction and air drag stay small enough to ignore.

Lineage

Christiaan Huygens's studies of pendulums in the 1650s and 1670s, aimed at building an accurate clock, established that a pendulum's period and its return height behave with a precision that begged for explanation. The dispute between Gottfried Leibniz, who insisted mass times velocity squared was the conserved "living force," and the Cartesian school, which tracked only mass times velocity, ran through the late seventeenth and early eighteenth centuries and was really an argument about what a conservation law for motion should even look like. The full resolution, recognizing potential and kinetic energy as two forms of one conserved total, and later extending that total to include heat, came together in the work of James Joule, Hermann von Helmholtz, and William Thomson in the middle of the nineteenth century, at which point mechanical energy conservation became the frictionless special case of a far more general law.

The strongest case for it

The payoff is that you can predict an outcome without simulating a process. A roller coaster designer can find the minimum height of a loop needed to keep a car on the track using only energy accounting, sidestepping a nightmare of changing curvature and changing forces. A ballistic problem, how fast a pendulum bob is moving at the bottom of its swing after being released from a given angle, becomes algebra rather than a differential equation. The same method scales up to a satellite trading potential energy for speed across an elliptical orbit, using the general form of gravitational potential energy in place of the near-surface version. Wherever friction and drag are genuinely small, this shortcut is exact, not approximate.

The strongest case against it

The law is conditional, and the condition is easy to forget. The moment a non-conservative force does real work, sliding friction, air drag, an inelastic deformation, a foot pushing off the ground, mechanical energy is no longer conserved on its own; some of it is converted into heat, sound, or permanent deformation that this accounting does not track. A common misconception is treating this as energy vanishing; it is not destroyed, only converted into a form outside the "mechanical" category, and the fully general law of energy conservation, which does include heat, still holds. A second misconception is applying the shortcut when an external agent adds energy to the system, a hand pushing a swing higher on each pass, without recognizing that this work must be added to the balance. The law also tells you nothing about timing: it relates two states, not how long it takes to get from one to the other.

Where it stands now

Mechanical energy conservation is broad-consensus physics, understood since the mid nineteenth century as the frictionless limiting case of the fully general conservation of energy. It remains in daily use exactly as derived here, and the boundary condition, no work done by non-conservative forces, is the honest and complete statement of when it applies.

Test yourself

A frictionless slide starts at a given height, dips into a valley below the starting level, then rises over a hump higher than the valley but lower than the start, before ending at a platform at some third height. Given only the starting height, the valley depth, the hump height, and the platform height, find the speed of a rider at the bottom of the valley, at the top of the hump, and arriving at the platform, using only energy accounting and no equations of motion along the curve. Then state the minimum starting height that would let the rider just barely maintain contact with the track at the top of the hump, and explain what additional piece of physics, beyond energy conservation alone, that minimum condition requires.