The block the water does not lift

Two identical blocks lie under a metre of water. One rests a hair above the floor of the tank, with a film of water beneath it. The other has been bedded down onto a smooth floor so perfectly that no water is underneath it at all. The first feels an upthrust of exactly the weight of the water it displaces. The second feels no upthrust whatever, and presses on the floor with its full weight plus the weight of the column of water above it.

That is a fact about apparatus rather than a paradox about physics, and the reason it is worth an essay is that the usual statement of Archimedes’ principle gives no hint that it could happen. “The upthrust equals the weight of the fluid displaced” contains nothing about the geometry of contact, and a rule whose failure mode is invisible in its own statement is a rule that has been taught in the wrong form.

Where the upthrust actually comes from #

A fluid at rest exerts pressure, and pressure acts perpendicular to whatever surface it touches, always pushing inward. There is no upward-pointing force anywhere in a tank of still water, and there is no tension in it either unless the water has been prepared with great care. What there is, is pressure that increases with depth, so the inward pushes on the bottom of a body are stronger than those on the top.

Written that way, the condition for Archimedes’ principle to hold becomes explicit: the fluid must touch the entire boundary of the body. The theorem is a divergence theorem, converting the surface integral of pressure into a volume integral of the pressure gradient, and a divergence theorem needs a closed surface. Seal one face against the floor and part of the boundary is no longer a fluid surface at all — it is a solid contact, carrying whatever force the contact carries, and the integral has nothing to say about it.

The same statement, seen from the other side #

The bedded block has a companion result that is much better known and is usually filed as a separate curiosity: the hydrostatic paradox.

The two results are the same result. In the paradox, the base carries a force that is not the weight of the fluid because the walls take up the difference. In the bedded block, the floor carries a force that is not reduced by an upthrust because there is no fluid underneath to supply one. Both are consequences of the fact that the pressure at a point knows the depth and nothing else — the same fact that makes a small force on a small piston lift a car — and that a resultant is an integral over a specified surface rather than a property of a volume.

The arithmetic, done once #

It is worth putting numbers on the two cases, because the difference is larger than the phrase “no upthrust” suggests.

Take a concrete block a metre on a side, density 2400 kg/m³, with its top face 1.2 m below the surface. Its weight is 23.5 kN. Held clear of the floor it displaces a cubic metre of water and feels an upthrust of 9.8 kN, so the floor carries 13.7 kN. Bedded down with no water underneath, the floor carries the block’s full 23.5 kN plus the weight of the water column above it, another 21.6 kN of pressure force on the top face — 45.1 kN in total, three and a third times as much.

The multiplier grows with depth without limit, because the top-face pressure grows and the block’s weight does not. At a hundred metres down the same block presses on a sealed bed with 1.0 MN, more than forty times its own weight, and none of that is a property of the concrete.

The arithmetic is worth doing once, and it is the other consequence of pressure being the same everywhere at one level. A hydraulic press and a sealed block are the same physics used two ways: in the press the pressure is transmitted to a piston and multiplies a force, and in the sealed block it is transmitted to every face except the one that is sealed off — and what is left is a downward push with nothing beneath it to push back.

Why the sides always cancel #

The pressure figure asserts that the side faces cancel exactly, and it is worth seeing that this is not a property of a rectangular block.

Take any submerged body at all and ask for the horizontal resultant of the pressure on it. Pressure at a point depends on depth alone, so consider a thin horizontal slice of the body: every point around its rim is at the same depth and therefore at the same pressure, and a constant pressure acting inward all round a closed curve has zero resultant. That is true of a circle, a crescent, a shape with holes in it — anything, provided the fluid reaches all the way round the slice. Stack the slices and the whole body’s horizontal force is zero.

Which is a stronger statement than it looks, and it is worth holding beside the vertical case. The vertical resultant depends on the shape of the body, through its volume; the horizontal resultant is zero regardless of shape, and depends on nothing. So the two directions fail differently when a seal is introduced: sealing the underside destroys the vertical result and leaves the horizontal one intact, while sealing a side face destroys the horizontal cancellation and produces a sideways force of exactly the kind that presses a suction cup against a wall.

It also settles a question about dams that is often asked backwards. The horizontal thrust on a dam depends only on the depth of water and the width of the wall, not on how much water is behind it — a reservoir a kilometre long and a pond ten metres long, at the same depth, push equally hard. The cancellation argument is the reason: everything except the wall itself is a closed rim at each depth.

Does it happen? #

The perfectly sealed case is an idealisation and the effect is not. Three real situations produce it to varying degrees.

A caisson or a gravity structure on a seabed. A large concrete base set down on soft sediment can develop a genuine seal, and the design load on it is then the full hydrostatic head rather than the buoyant weight — a difference of thousands of tonnes on a structure of any size. Offshore practice puts skirts and drainage under such bases specifically so that the water pressure has a route underneath, because the alternative is to design the foundation for a load that does not exist if the seal fails and does exist if it holds.

A ship aground on a flat mud bottom. The suction that has to be overcome to refloat a grounded vessel is often far larger than its weight, and the cause is precisely this: the mud has sealed the hull and the water above is pressing down with no counterpart below. Salvage crews break the seal by jetting water under the hull before pulling.

A suction cup, which is the same thing upside down. Everything in a suction cup’s grip is atmospheric pressure on one face and nothing on the other, and the reason it fails on a rough surface is that a leak path lets the pressure equalise. A bedded block is a suction cup that nobody deliberately made.

Whether a block bedded on a floor develops a genuine seal is a wetting question as much as a mechanical one. The same volume of liquid sits quite differently on different solids, and the contact between block and floor has to exclude liquid completely for the effect to appear at all. On most real surfaces it does not: there is roughness, there is a film, and the water gets underneath. That is why the demonstration is harder to arrange than the argument suggests.

How to demonstrate it, and how the demonstration usually fails #

The experiment is easy to describe and hard to do, which is itself informative about the mechanism.

Take a flat-bottomed block of something denser than water — a machined aluminium slug is ideal — and a sheet of plate glass. Wet both, press the block down with a slight twisting motion to expel the film, and lower the assembly into a tank. If the seal has taken, the block cannot be lifted by an upward force equal to its buoyant weight; it needs the full weight plus the head of water above it. Slide it sideways instead, breaking the seal at an edge, and it becomes light immediately.

What usually goes wrong is that the seal never forms, and there are two distinct reasons. The first is roughness: any groove connecting the underside to the surrounding water is a channel through which the pressure equalises, and a channel a micrometre deep is enough given time. The second is that water wets both surfaces, so capillary action actively draws it into any gap rather than excluding it — which is why the experiment is easier with a hydrophobic block and why greasing the glass makes it work.

The failure is therefore not a failure of the physics but of the boundary condition, and being able to say which is the whole benefit of having derived the principle from a surface integral rather than having remembered it.

What a floating body is doing #

The floating case is worth revisiting with the surface-integral view in hand, because “floats when it is less dense” turns out to be a special case too.

The comparison of densities is a summary that holds when the only surfaces involved are the wetted hull and the free atmosphere — and even then it says only whether the body floats, not how it sits, which is a question about where the buoyant resultant acts relative to the weight. A steel needle floating on water is not less dense than water; it is held by the surface, which is not a skin and behaves like one, bearing a load the pressure integral does not contain. A block resting on a wet floor is not denser than water; it is sealed.

A contact line — where solid, liquid and gas meet — is exactly the object that decides whether a body’s boundary is sealed. It is also notoriously badly behaved: it pins on irregularities, it moves in jumps rather than smoothly, and its position depends on the history of the surface as much as on the materials. So the question of whether a block is sealed has no clean answer, which is the honest reason this effect is quoted more often than it is demonstrated.

Weighing, and the correction nobody can avoid #

The remark that air buoyancy matters for a precision mass measurement and for nothing else is worth a number, because the number is large enough to have shaped how the kilogram was defined.

Air has a density near 1.2 kilograms per cubic metre. A kilogram of water occupies a litre and displaces 1.2 grams of air; a kilogram of platinum–iridium occupies 47 cubic centimetres and displaces 0.056 grams. Put the two on a balance in air and they do not balance: the water is lighter by a gram and a bit, from buoyancy alone, and no amount of care with the balance recovers it.

Even between two metals the effect is not small. A stainless-steel kilogram and a platinum–iridium one differ in volume by about eighty cubic centimetres, so their air buoyancies differ by roughly ninety-five milligrams — against a comparison uncertainty that the best balances push below a microgram. The buoyancy correction is therefore five orders of magnitude larger than the measurement’s own precision, and everything about the accuracy of a mass comparison comes down to how well the density of the air in the room is known on the day.

Which is why the correction is computed rather than eliminated: from the pressure, the temperature and the humidity, through an equation of state for air, to a density good to a part in ten thousand. And why standards of nominally identical material and volume are preferred wherever possible, since two bodies of the same volume have the same buoyancy and the correction cancels out of their difference without ever being computed.

The connection to the rest of this essay is exact. Every one of those corrections is a surface integral of pressure over a fully wetted boundary, and each of them would fail in the same way if part of the standard were sealed to its support. Mass metrologists handle their artefacts with tongs on a narrow contact for reasons of contamination, and the effect described here is a second one.

How long a seal lasts #

The transience noted above deserves a mechanism, because “the water gets in eventually” is not a timescale.

What has to happen is that the pressure under the block rises from atmospheric to hydrostatic, and it does so by fluid flowing in through whatever gap there is. Flow through a narrow gap is driven by the pressure difference and resisted by viscosity, and the rate falls as the cube of the gap’s height — so a gap of a micrometre passes a thousandth of what a gap of ten micrometres passes.

The consequence is a very steep dependence on surface finish. A block lapped flat against polished glass can hold for minutes; the same block on a ground surface equalises in a second or two, and on anything rough it never seals at all. There is no characteristic time for the phenomenon, only a characteristic time for a particular pair of surfaces, and it varies over orders of magnitude with a quantity nobody measured.

On a seabed the same argument runs with sediment in place of a gap, and the resisting quantity is the sediment’s permeability. Fine silt and clay are enormously less permeable than sand, so a structure bedded on clay can retain a pressure deficit underneath it for hours or days while the same structure on sand equalises in minutes. That is why the presence or absence of the effect in practice is decided by the geology rather than by the structure — and why the same design is safe on one site and not on another.

Where the model stops #

A perfect seal is unattainable and its absence is gradual. A real block on a real floor has some fraction of its underside in contact and the rest wetted, and the upthrust is the same fraction of Archimedes’ value. The two extreme cases are the limits of a continuum, and where a given case sits on it is a surface-roughness question with no clean answer.

A seal does not last. Water is a fluid and will find a path: over minutes or hours it seeps into a nominally sealed contact, the pressure underneath rises toward hydrostatic, and the upthrust reappears. The effect is transient in almost every laboratory demonstration of it, which is why the demonstration usually uses a smooth block on glass with a film of grease and is over in seconds.

Air is a fluid too, and the same argument applies to it. Every object in a room is subject to an atmospheric upthrust equal to the weight of the air it displaces, which for a person is about a newton — enough to matter for a precision mass measurement and for nothing else. A body sealed to a surface in air is subject to the same failure of the principle, and it is the mechanism of a suction cup rather than a curiosity.

And the pressure field is assumed hydrostatic. Everything here treats the water as still. A body on a seabed under a passing wave has a pressure field that varies in time, and the pressure underneath — which has to diffuse in through the sediment — lags the pressure above, so the net force oscillates and can pump the body out of the bed. That is a real failure mode for pipelines and it is a flow problem rather than a statics one, belonging to the collection that owns flows.

What the pictures cannot show #

The hero figure draws two blocks side by side, and the thing that distinguishes them — the presence or absence of a film of water a few micrometres thick — cannot be drawn at any scale that also shows the tank. The whole effect turns on a gap that is invisible in the figure and is the only difference between the two cases.

Nor can any of these figures show a resultant. Pressure arrows are drawn on faces, and the single upward arrow of buoyancy is not a force acting anywhere — it is a sum, drawn at a point chosen for convenience, and the choice of point matters for torques even though it does not for forces. A ship’s righting moment is entirely a question about where that fictitious arrow is placed.

Where this ladder goes next #

The ladder began with the weight of the water that is not there, which is Archimedes stated as a substitution argument, and went on to stability and to the depth past which a compressible body must sink. This rung goes back to the beginning and asks what the principle is a theorem about, and the answer — a closed surface in a fluid — is where its exceptions come from.

The habit worth carrying away is a discipline about principles that summarise integrals. Ask what surface the integral was taken over, and check that the surface is still there. Archimedes assumes a fully wetted boundary; Gauss’s law assumes a closed one; the work-energy theorem assumes a path that the body actually took. Each of them is stated in a form that hides its own assumption, and each has a well-known “paradox” attached to it that is nothing more than the assumption failing quietly.

What is left on this ladder is the body that is not in one fluid — a block straddling an interface, or floating on a stratified column — where the displaced volume has to be counted twice with two densities, and where the stability question changes character entirely.