The balloon that leans the wrong way

A car pulls away from traffic lights with the windows shut. A bag on the back seat slides backward, a pendant hanging from the mirror swings backward, the passengers are pressed back into their seats — and a helium balloon a child is holding on a string swings forward, towards the windscreen, and stays leaning forward for as long as the car keeps accelerating. When the driver brakes, it leans back. It looks like a trick, and it is the most direct demonstration there is that buoyancy is not a force that points “up”. It points against gravity, and inside the car gravity no longer points down.

Gravity with a sideways part #

Archimedes’ principle is built out of the pressures on a body’s faces: the fluid pushes harder on the bottom than on the top because the bottom is deeper, and the difference is exactly the weight of the fluid displaced. That derivation used gravity in one place only — to say why pressure increases downward. Change what makes the pressure vary and the buoyant force changes with it, automatically and with no new physics.

In a frame that accelerates at $\mathbf{a}$, every object behaves as though gravity had been replaced by

$$\mathbf{g}_{\text{eff}} = \mathbf{g} - \mathbf{a}.$$

That is the content of the equivalence principle read the other way round: an acceleration cannot be distinguished from a gravitational field, so it can be treated as one. The fluid in the cabin is at rest relative to the cabin, which means every parcel of it is being accelerated forward along with the car, and the only thing that can accelerate a parcel of fluid is a difference in pressure across it. So the pressure in the cabin increases backward as well as downward, and the direction in which it increases fastest is along $\mathbf{g}_{\text{eff}}$. The surfaces of constant pressure are tilted to lie perpendicular to it. The free surface of the water is one of those surfaces, which is why liquid in an accelerating tanker settles at a slope, and the air above the water has tilted isobars of its own, much more widely spaced because air is so much lighter.

Put a body in that fluid and Archimedes’ argument goes through word for word, with $\mathbf{g}_{\text{eff}}$ in place of $\mathbf{g}$. The pressure is higher on the body’s back-and-lower side than on its front-and-upper side, and the net force from the fluid is the weight of the displaced fluid measured in the effective gravity, directed along $-\mathbf{g}_{\text{eff}}$: up and forward. A body denser than the fluid feels its own effective weight win and moves back and down along $\mathbf{g}_{\text{eff}}$. A body lighter than the fluid moves forward and up along $-\mathbf{g}_{\text{eff}}$. The balloon is not leaning the wrong way. It is doing exactly what a cork does in a bucket, in a gravity that has been rotated.

The forces, with numbers #

A free-body diagram makes the equality of the angles concrete, and it shows that the balloon’s lean is not a fluke of any particular mass or size.

The balloon’s buoyancy is large compared with its weight — the displaced air weighs three times as much as the helium and the rubber together — and it points up and forward. For the balloon to accelerate with the car, the sum of buoyancy, weight and string tension must be its own mass times the car’s acceleration, a small forward force. Solve for the tension and it comes out pointing backward and down, along the string, and the string’s angle from the vertical is exactly $\arctan(a/g)$, the same tilt as $\mathbf{g}_{\text{eff}}$. That equality is not approximate. The balloon’s net buoyancy-minus-weight is a force along $-\mathbf{g}_{\text{eff}}$, the string must supply the rest, and the balance is independent of the balloon’s size, the helium’s density and the rubber’s mass. Any lighter-than-air body on a string leans forward by the same angle.

The plumb bob has no buoyancy worth mentioning — the air it displaces weighs a thousandth of what it does — so its string must provide the whole of the forward force, and it hangs back by the same angle. The two instruments lean in opposite directions along one line, and the line is the direction of the effective gravity. A child with a balloon in a car is holding an accelerometer that reads the car’s acceleration as an angle: 17 degrees for 3 metres per second squared, and 39 degrees for the eight metres per second squared of an emergency stop.

The explanation often offered is that the air in the car “sloshes backward” when it accelerates and crowds the balloon forward. There is a transient slosh when the acceleration begins, but it dies away in a fraction of a second, and the balloon keeps leaning for as long as the acceleration lasts. What holds it forward is the steady pressure gradient in air that is at rest relative to the car — and that gradient exists only because the car is closed. Open the windows at speed and the air in the cabin is no longer simply carried along; the effect becomes a mess of wind.

Braking reverses everything at once #

The rule is symmetric in a way that makes a good test of whether it has been understood.

When the car brakes, $\mathbf{a}$ points backward, so $\mathbf{g}_{\text{eff}}$ tilts forward. Every dense object swings forward, as the passengers do, and the balloon swings back. On a bend the effective gravity tilts outward, away from the centre of the turn, so the passengers lean out and the balloon leans in — into the turn, the way a cyclist leans. That is the origin of the essay’s title as it is usually observed: a balloon in a car going round a corner leans towards the inside of the corner while everything else leans out.

The same tilt governs anything that floats and can heel. A ship turning hard sits in an effective gravity tilted outward from the turn, and its righting moment is measured against that direction rather than against the true vertical: the hull heels outward until its centre of buoyancy lies along the tilted gravity through its centre of mass, exactly as it would on a sloping sea with no turn at all. A passenger on deck feels the floor tilt and feels no sideways force — the floor has rotated to be perpendicular to the gravity everyone aboard is standing in. Nothing about the pressure at a point changes either: it is still the same in every direction, because the argument that makes it so compares forces that scale with a small volume’s surface against forces that scale with its volume, and replacing $\mathbf{g}$ by $\mathbf{g}_{\text{eff}}$ changes only the second kind. What tilts is the direction in which pressure grows, and the depth it depends on becomes a depth measured along $\mathbf{g}_{\text{eff}}$.

A lift cannot change how deep a boat floats #

The effective gravity changes size as well as direction, and here the rule makes a prediction that sounds wrong until it is worked through.

In a lift accelerating upward at one gravity the block weighs twice as much. It would seem obvious that it must sink deeper to find twice the buoyancy. It does not, because the water weighs twice as much too, and the pressure at every depth has doubled: the same draught displaces a volume of water whose effective weight has doubled along with the block’s. The balance is between two forces that both carry the factor $g + a$, so the factor cancels, and the depth at which a body floats is set by the ratio of its density to the liquid’s and by nothing else.

Two things follow. A ship’s waterline does not change as it rides up the face of a large wave and down the back, although the effective gravity changes by tens of per cent through the motion — the draught marks on a hull are a measure of the load, not of the sea. And a hydrometer, which measures a liquid’s density by how deep it floats, reads the same in a lift, on the Moon or in a centrifuge, because it is measuring a ratio of densities and the gravity is in both halves.

What does change is everything that is not a ratio. The pressure at the bottom of the tank doubles. So does the pressure difference that makes the force on a submerged body, and with it the stiffness with which a floating body resists being pushed down — so a floating block in an upward-accelerating lift bobs faster when disturbed, by the square root of the factor, even though it floats at exactly the same level.

Free fall abolishes floating #

At $a = -g$ — a lift whose cable has snapped, a spacecraft in orbit, an aircraft flying a parabola — the effective gravity is zero. The right-hand side of every hydrostatic equation vanishes.

The pressure in the liquid is then uniform. There is no pressure difference between the top and bottom of any body in it, and so no buoyant force, whatever the body’s density. The block in the lift figure is in equilibrium at every depth: the force balance reads nought equals nought. A bubble in a glass of water in orbit stays where it was put. So does a grain of sand. A candle flame, whose hot gas normally rises because it is less dense than the air around it, has no reason to rise; it becomes a faint blue sphere, fed only by the slow diffusion of oxygen inwards, and it often goes out.

What takes over when buoyancy leaves is surface tension, and the handover can be put on a scale. A drop or a bubble keeps the shape its surface tension gives it when the difference in hydrostatic pressure across it is small compared with the pressure its curved surface supports. The length at which the two are equal is the capillary length, $\ell_c = \sqrt{\gamma/\rho g_{\text{eff}}}$, and it depends on the effective gravity.

On the Earth only things smaller than a few millimetres are governed by surface tension — dewdrops, the meniscus in a tube, a water strider’s footprint. In orbit, everything up to a few metres is: a litre of water released from a bag becomes a wobbling sphere, and a bubble inside it has no preferred place to go. The physics of liquids in a spacecraft’s fuel tank is capillary physics at the scale of a room, and fuel has to be herded towards the outlet by vanes and screens that exploit surface tension, since nothing else will move it.

A centrifuge is buoyancy pointing inward #

The same rule applied in a rotating frame gives the most useful machine in the subject.

In a frame turning with the liquid, each parcel feels an outward centrifugal acceleration $\omega^2 r$, and the effective gravity is the vector sum of that and the ordinary downward $g$. For a tube spinning at thousands of revolutions a minute the outward part dwarfs the downward one, and “down” means “away from the axis”. The pressure rises outward, the isobars are circles, and Archimedes’ principle does the rest. A denser particle is heavier than the liquid it displaces in this gravity and moves outward, which is how blood is separated into its cells and its plasma and how a laboratory collects a pellet of cells at the bottom of a tube. A lighter body is buoyed inward. Oil droplets in a cream separator, air bubbles in a spun bottle, and a cork in a bucket swung round on a rope all move towards the axis — towards what is, in this frame, up.

The factor by which a centrifuge multiplies gravity multiplies the speed at which particles separate, since that speed is set by the effective weight against the liquid’s drag. A factor of eight hundred turns a separation that would take a week under gravity into one that takes a quarter of an hour. And the paraboloid a spinning liquid’s surface forms is the same statement for the free surface: it is an isobar, perpendicular everywhere to an effective gravity that tilts further outward the further from the axis it is.

Where the frame picture stops #

The acceleration is steady. Every figure assumes the cabin, the lift or the centrifuge has been accelerating long enough for the fluid inside to come to rest relative to it. When the acceleration changes, the fluid lags, sloshes and oscillates, and a balloon in a car that pulls away jerkily swings past its equilibrium angle and back before settling. The time it takes to settle is set by the fluid’s own waves, not by the hydrostatics.

The frame accelerates as a whole. In a rotating frame the effective gravity varies from place to place, and a body large enough to span a range of radii feels a range of effective gravities. More importantly, a body moving through a rotating fluid feels the Coriolis force as well, which the static picture leaves out entirely and which deflects a rising bubble sideways.

The fluid is incompressible for the purpose of the force. Air in a car is compressible, but its density changes by a negligible fraction across a cabin. In a gas centrifuge, where the effective gravity reaches hundreds of thousands of $g$, the gas at the wall is many times denser than at the axis, and the rule has to be applied with the density a function of radius — which is precisely what makes such centrifuges separate isotopes.

Free fall is never perfect. A spacecraft in orbit carries residual accelerations from drag, attitude control and the tidal gradient across its own length, of order a millionth of $g$ — which is why the capillary-length figure marks orbit there rather than at zero. On the scale of a fuel tank those residuals are what decide where the liquid ends up after weeks of coasting.

The slosh before the surface settles #

The tilted surface in the first figure is drawn with the water at rest relative to the tank, which is the end of a process the picture does not contain. When the car starts, the water’s surface does not tilt smoothly into place; it overshoots and rocks at the tank’s natural sloshing frequency, and in a tanker lorry that sloshing is the hazard, not the steady tilt. Baffles inside the tank do nothing to the final slope and everything to the rocking.

Nor does a single frame show that the balloon is a measuring instrument with a response time. It swings to its angle over a second or so, damped by the air it is moving through, and a car that accelerates for less than that time never shows the full lean at all.

Still open: how liquids arrange themselves when nothing is pulling them #

For a spacecraft carrying cryogenic propellant, the question of where the liquid is at the moment an engine must restart is a matter of life and death, and it is not fully predictable. With the effective gravity reduced to residual millionths, the liquid’s position in the tank is decided by a competition among surface tension, the tank’s geometry, the wetting of its walls, heat leaking in and boiling the liquid at the warmest surfaces, and the small accelerations of the spacecraft itself. Each is understood alone; together they produce behaviour that numerical models struggle to reproduce over the weeks of a long mission. Experiments in orbit and in drop towers continue to find configurations that the models did not predict, and the problem of storing and transferring liquid hydrogen and oxygen in orbit — central to any refuelling of spacecraft — is still being solved partly by trial.

The question the frame picture leaves next is what happens when the fluid is not at rest in the accelerating frame: a rotating liquid with a disturbance in it, where the Coriolis force enters and the effective gravity is no longer the whole story. The habit worth carrying from here is to find the direction of the effective gravity before asking which way anything will move. Buoyancy points against gravity, whatever gravity happens to be in the frame doing the asking — and in any accelerating vehicle, the lightest thing in it is the one that points at where the vehicle is going.