Where the upward force comes from
At its defaults it draws where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.
buoyancy is one function in lib/figures/fluids.js —
matter that will not hold a shape, and the forces in it. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.
A heeled hull, and the couple it makes
The options are the ones Slide or topple passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A rectangular hull of beam 3 m heeled 16°, with the waterline solved so that it displaces the same volume it did upright. The centre of buoyancy has moved 0.215 m to the low side, and weight and buoyancy now act along two lines 0.050 m apart — a couple that turns the hull back upright.
The height at which hot air stops being light enough
The options are the ones The balloon a cold morning lifts passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.
What a cubic metre of hot air can lift
The options are the ones The balloon a cold morning lifts passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The lift of each cubic metre of an open balloon's air, in kilograms, against the temperature inside it, at sea-level pressure on a morning at −10, 15 and 35 °C. The inside and outside are at the same pressure, so the lift is the outside density times one minus the ratio of absolute temperatures. At the fabric's working limit of about 120 °C (shaded beyond) a cubic metre lifts 0.444 kg on the cold morning, 0.327 at 15 °C and 0.248 at 35 °C. A cubic metre of helium lifts 1.06 kg (dotted) — three to five times as much — which is why a hot-air balloon must be so large: a four-person balloon holds about 2800 m³, the volume of a house.
Why balloons fly in the cold of the morning
The options are the ones The balloon a cold morning lifts passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
What a 2800 m³ balloon can carry beyond its own envelope — basket, burners, fuel and people — against the outside temperature, at sea-level pressure, with the inside held at 100 °C and at the fabric's limit of 120 °C. Cold air is denser and makes the ratio of temperatures more favourable at the same time, so the useful load falls steeply as the day warms: at 120 °C inside, 980 kg at 0 °C, 733 kg at 20 °C and 569 kg at 35 °C. Balloonists fly at dawn mainly because the air is calm then, before the sun starts convection; but the dawn air also lets the same envelope carry hundreds of kilograms more, or the same load with the envelope much cooler, which preserves the fabric and saves fuel.
Bigger envelopes fly higher, because the load is shared
The options are the ones The balloon a cold morning lifts passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The ceiling of a hot-air balloon at 120 °C against its envelope volume, on a logarithmic axis, for payloads of 300, 550 and 1,500 kg, with the envelope's own weight (0.13 kg/m², fabric with its load tapes and fittings) counted. Lift grows with the volume and the fabric only with the area, so a larger balloon carries its fixed load more cheaply and climbs higher: for 550 kg, 6.9 km at 3,000 m³, 15.2 km at 10,000 and 24.6 km at 100,000. The altitude record for a hot-air balloon, 21,027 m in 2005, used an envelope of about 45,000 m³ carrying a pressurised gondola; this calculation gives such an envelope a ceiling of 18.4 km with 1.5 tonnes aboard, and the burners had to be kept alight in air too thin for an ordinary flame.
What checks it
physicscheck asserts something about buoyancy that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Slide or topple
Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.
FluidsThe balloon a cold morning lifts
A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.
FluidsThe balloon that floats at a density
A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.
FluidsThe balloon that leans the wrong way
When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.
FluidsThe block the water does not lift
A block bedded flat on the bottom of a tank, with no water underneath it, feels no upthrust at all. It is fully submerged, Archimedes' principle is not suspended, and it presses on the floor with more than its own weight — because buoyancy is not something a fluid has, it is what the bottom face is doing, and a face the water cannot reach does nothing.
FluidsThe body that displaces two things
Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.
FluidsThe depth past which it must sink
A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.
FluidsThe floating ice that still raises the sea
An ice cube melting in a glass of water leaves the level exactly where it was, and the reason is exact: a floating body displaces its own weight, and the meltwater has that weight. The same argument is often used to say that melting sea ice and ice shelves cannot raise the sea. It nearly works. The ice was floating in salt water and melts into fresh, which is lighter, so the meltwater overfills the hole by almost three per cent — and the ice that really fills the sea is the part of a grounded glacier above the thickness at which it would float.
FluidsThe grounding line with nowhere to stop
Where an ice sheet flowing into the sea lifts off its bed and begins to float, buoyancy fixes how thick it is: just thick enough to touch the rock, about 1.1 times the depth of the water. The ice crossing that line goes as the 19/4 power of the thickness, so twice the depth sends out twenty-seven times the ice. On a bed that deepens inland, a grounding line that retreats a little moves into deeper water, exports more, and retreats further. No steady state can sit there, and an ice sheet whose edge reaches such a slope crosses the whole of it.
FluidsThe log that floats on its corner
A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.
FluidsThe water the dent displaces
A steel needle laid carefully on water floats, and the usual explanation is that surface tension holds it up instead of buoyancy, as though the two were rival mechanisms. They are not rivals. The needle presses a dent into the surface, and the upward pull of surface tension along its edges is exactly the weight of the water missing from that dent. Archimedes' principle holds without amendment: a floating needle displaces its own weight of water. Most of that water is simply not underneath it.
FluidsThe weight of the water that is not there
A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.
FluidsWhy a ship comes back upright
Whether a floating body rights itself or rolls over is not decided by its weight, its density or how deep it sits. It is decided by the shape of the slice the water cuts through it, and the number that settles it can be worked out before the vessel is built.