Balance
Why Things Stand Up, and Why They Fall Over
One Point Carries the Whole
Balance a broom on one finger and you will find a single spot, well toward the brush, where it rests. Put the finger anywhere else and the broom tips. That spot is the centre of mass: the one point at which the whole weight of an object can be treated as acting, however its material is spread. For a uniform ruler it is the middle. For a broom it is near the heavy end. For a standing person it is a little below the navel, and it moves when you raise an arm.
The centre of mass earns its name by making a complicated body simple. Throw a wrench spinning across a room and every part of it traces a tangle, but one point traces a clean arc, the same one a thrown pebble would trace. Gravity acts on the object as if all of it were gathered there. The rest is rotation about that point, and the Rotation page handles that half. This page is about the point itself, and the one rule that follows from it.
The Rule of the Base
An object resting on the ground stands as long as its centre of mass is above its footprint, the patch of ground its support covers. Tilt it, and the centre of mass swings to one side. As long as a plumb line dropped from it still lands inside the footprint, the ground pushes back in a way that rights the object. The moment the line crosses the edge, the same push becomes a lever that tips it over. There is no gradual failure. The rule is a line on the floor.
That single rule sets the shape of everything that needs to stand. Wide and low is stable; tall and narrow is not. A racing car sits a hand’s breadth from the ground with its wheels far apart because a car that corners hard is trying to swing its centre of mass past the outside tyres. A tall lorry rolls where a car does not. A climber leans in toward the rock and drops a hip to bring the centre of mass over the feet. The Leaning Tower of Pisa leans by about four degrees and stands, because the plumb line from its centre of mass still lands inside its base. It leaned by five and a half before engineers pulled it back in the 1990s, with the line uncomfortably close to the edge.
The angle at which a thing tips is fixed by nothing but the width of its base and the height of its centre of mass. A box twice as tall as it is wide tips at about twenty-seven degrees. A box as wide as it is tall tips at forty-five. A short wide one hardly tips at all. The picture below lets you tilt the floor and watch the plumb line find the edge.
Stable, Unstable, and Neutral
There are three ways to rest. A ball at the bottom of a bowl is stable: nudge it and it rolls back. A ball balanced on top of an upturned bowl is unstable: nudge it and it leaves. A ball on a flat table is neutral: nudge it and it stays wherever it stops. The difference is whether a small displacement raises the centre of mass, lowers it, or leaves it where it was, and everything that stands is one of the three.
A pencil balanced on its point is the unstable case, and it shows how hopeless unstable really is. Any error in the placement, however small, grows exponentially, doubling about every fifteenth of a second for an ordinary pencil. Even a placement good to the width of an atom would hold for only about two seconds before the pencil was visibly leaning. No hand is that steady and no table is that still. Nothing balances on a point for long, not because of clumsiness, but because the physics amplifies whatever clumsiness there is.
Standing Is Falling, Caught in Time
A person standing upright is the pencil. The centre of mass sits about a metre above a footprint the size of two shoes, and the body is in the unstable case: left alone, it would be on the floor in about a second. It does not fall because it never stands still. Sensors in the feet, the inner ear and the muscles report the lean, and the ankles and hips push back, several times a second, so the body sways inside a circle a centimetre or two across without its owner noticing. Standing is a continuous act of catching. That is why standing for hours is tiring, and why it takes a baby a year to learn.
Walking is the same act stretched out. At each step the body leans past its footprint and falls forward, and the swinging leg lands in time to make a new base under the fall. Running is walking with the catching arriving later. A bicycle at speed does the trick sideways: as the machine starts to fall to one side, the geometry of its front fork steers the wheel toward the fall and brings the base back under the rider. It was long taught that the spinning wheels keep a bicycle up the way a gyroscope stays up. In 2011 a team in Delft built a bicycle with no gyroscopic effect and no steering trail at all, and it balanced itself anyway. The steering does most of the work. The spin is a bonus.
A tightrope walker’s pole works on both halves of the problem. It droops below the wire at its ends, which lowers the walker’s overall centre of mass toward the rope. And its length makes the walker slow to rotate, which buys time for the catching. The pole does not add stability. It adds seconds.
Standing on Compression
A wall stands because every stone presses down on the one below and nothing asks the stones to bend. Stone is strong when squeezed and weak when stretched, and a flat stone beam across a doorway fails within a few metres because its underside is being stretched. The arch is the answer, and it is a balance problem turned into a shape. Each block leans on its neighbours; the weight is turned sideways, block by block, and arrives at the ground as a push along the curve. Nothing in a true arch is stretched. Everything is squeezed.
The ideal curve is the one a hanging chain makes, turned upside down. A chain hangs entirely in tension, and the same curve stood on its head is entirely in compression. Robert Hooke saw this in 1675, and Antoni Gaudí designed churches with hanging models of string and small weights, photographed and inverted. The Gothic cathedral’s flying buttresses exist because a pointed arch pushes outward at its feet, and something has to lean back against that push. An egg is strong for the same reason a dome is: press on it and the shell is squeezed everywhere and stretched nowhere.
The Stack That Leans Out
Take a stack of identical blocks and slide the top one out as far as it will go. It stands until its centre is exactly over the edge of the block below: half a block length. Now slide the top two together. Their shared centre of mass is a quarter of a block from the edge of the third. The top three together can overhang the fourth by a sixth. Adding blocks, each new one at the bottom, the total overhang is a half plus a quarter plus a sixth plus an eighth and so on: one over twice the count, for every block in the stack.
That series grows without limit, though slowly. Four blocks can reach a full block length past the table. Thirty-one are needed for two block lengths, and about two hundred and thirty for three. In principle there is no overhang that enough blocks could not reach. In practice the blocks are never perfectly flat and the tower is never perfectly still. The picture below stacks them, and marks where each plumb line lands.
Where Balance Stops Being Statics
Everything above is about things at rest. Set them moving and the rule of the base stops being the whole story. A spinning top stands on a point no pencil could, because its spin turns a fall into a slow circling, which the Rotation page explains. A ship floats stably with its centre of mass above the centre of the water it displaces, which the rule of the base would seem to forbid, because a tilting hull shifts the buoyancy sideways faster than it shifts the mass; the Pressure page has the buoyancy. A kite held up by the wind, a plane in level flight, a cyclist on a bend: each is held by a balance of forces in motion, and each is stable only while the motion lasts.
At the smallest scales balance has a different enemy. A dust grain cannot be balanced on a point at all, because the air molecules that strike it deliver kicks comparable to its weight, and the Noise page describes that hammering. Balance, in the end, is a statement about a world where the disturbances are small compared with the thing disturbed. That is true of towers and cathedrals and people. It stops being true somewhere around the size of a grain of pollen, and below that nothing stands still at all.
