Buoyancy Simulator

What buoyancy is

Buoyancy, or upthrust, is the upward force a fluid exerts on anything in it, and it equals the weight of the fluid the object pushes aside. That is Archimedes’ principle, and as a formula it is F_b = ρ_f × V_sub × g: the density of the fluid times the volume under the surface times the gravitational field strength. What the object is made of does not appear. Only how much of it is under, and what it is under, decides the force.

The force comes from pressure. Pressure in a liquid grows with depth, as P = ρgh, so the liquid presses harder on the bottom of a block than on its top. For a block with flat faces the difference is the pressure difference across its height, ρ_f g h, times the area of its base, and height times base area is the block’s volume. Pressure itself is force per area, which the pressure calculator works out from P = F/A.

How to use the simulator

Play lowers the block on a spring balance into a can filled to its spout. As it goes in, the liquid it pushes aside runs out of the spout into the beaker, the buoyant force grows, and the balance reading falls by exactly as much, because the balance and the liquid share the block’s weight between them. The arrows on the block show the three forces to one scale, so the two upward ones always add up to the downward one. Drag the scene, use the arrow keys or move the slider under it to hold the block at any depth, and choose a material and a liquid or set either density yourself.

The plot shows the whole run at once. The buoyant force rises in a straight line while the block goes in and then stays level, and the beaker shows why: the liquid caught weighs the same as the buoyant force at every depth. A spring balance measures a force by how far its spring stretches, which is Hooke’s law, and the Hooke’s law calculator works that relationship.

Worked example: an aluminium block in water

The simulator opens on a 100 cm³ block of aluminium, density 2700 kg/m³, lowered into water at 20 °C, density 998 kg/m³, with g = 9.81 m/s².

• Mass 2700 × 0.0001 = 0.27 kg, so in air it weighs 0.27 × 9.81 = 2.65 N.
• Fully under, it displaces 100 cm³ of water, whose mass is 998 × 0.0001 = 0.0998 kg. The buoyant force is that water’s weight, 0.0998 × 9.81 = 0.979 N.
• The balance carries the rest: 2.65 − 0.979 = 1.67 N, which is the block’s apparent weight in water.
• Half-way in, half the volume is displaced, so the buoyant force is 0.49 N and the balance reads 2.16 N.

The block sinks because 2700 kg/m³ is more than 998, and its size cannot change that. Double the volume and every force doubles, but the water still takes 998 ÷ 2700, or 37 percent, of the weight. The weight itself is mass times g, the same sum the force calculator does for F = ma with the acceleration set to g.

When it floats, and how much is under

A block less dense than the liquid goes in only until the buoyant force has grown to its whole weight. There it floats, the string goes slack and the balance reads zero. The fraction under the surface is then the ratio of the densities, V_sub/V = ρ_block/ρ_liquid, because the liquid displaced has to weigh the same as the block. Choose ice in seawater: 917 ÷ 1025 = 0.895, so 89.5 percent of the ice is under, and the beaker catches 89.5 mL of seawater, which has the same mass as the ice, 91.7 g. Holding the ice right under would take an extra push of 0.106 N. In fresh water the same ice floats lower, with 91.9 percent under, and in ethanol, at 789 kg/m³, it sinks.

Density is all that decides it, which is why iron floats on mercury. At 7874 kg/m³ against 13,534 it floats with 58.2 percent under, lead floats with 83.8 percent under, and gold, at 19,282, sinks. A steel ship floats for the same reason: what counts is the average density of the hull and the air inside it, which is lower than that of water.

Density from two weighings

The same sums run backwards measure density without a ruler. Weigh an object in air and again hanging in water, and the loss of weight is the buoyant force, the weight of water with the object’s own volume. So the object’s density is the water’s density times its weight in air divided by that loss: ρ = ρ_water × W ÷ (W − W_water). For the aluminium block, 998 × 2.65 ÷ (2.65 − 1.67) gives 2700 kg/m³ to three figures. This is Proposition 7 of Archimedes’ On Floating Bodies, and T. L. Heath, in his 1897 translation, took it to be how Archimedes tested the famous crown. The density calculator finds density the direct way, from a mass and a volume.

Does gravity change anything?

The forces, yes; whether it floats, no. Weight and buoyant force are both proportional to g, so at the Moon’s 1.62 m/s² every force here falls to about a sixth of its value on Earth, while the fraction under the surface stays exactly where it was: ice in water still floats with 91.9 percent under. What changes is how little force it takes to push the ice right under.

Common mistakes

• Deciding by weight. Heavy things do not sink because they are heavy; they sink because they are denser than the liquid. A tonne of cork floats and a grain of sand sinks.
• Putting the object’s density into the formula. F_b = ρ_f V_sub g uses the density of the fluid. The object’s own density only enters through its weight.
• Using the whole volume for a floating object. Only the part under the surface displaces liquid. A floating object’s buoyant force is simply its weight, and the volume under follows from that.
• Expecting the buoyant force to keep rising with depth. It grows only while more of the object is going under. Once it is fully under, the top and the bottom are both pushed harder by the same amount, so the difference stays put: the flat part of the plot.
• Mixing cubic centimetres and cubic metres. 1 cm³ is 10⁻⁶ m³. Put a 100 cm³ block into the formula as 100 and the buoyant force comes out a million times too big.
• Reading apparent weight as lost mass. The balance reads less because the liquid holds up part of the weight. The mass is unchanged, and lifting the block out brings the full reading back.

What this simulator leaves out

It is quasi-static: the block is lowered slowly and held at each depth, so nothing is ever in motion. Let a real block go and it accelerates, overshoots and bobs, and then drag and the liquid it carries along with it matter too. Surface tension is left out too, which is why a steel pin laid gently on water can rest there although steel is far denser: that is the surface holding it up, not buoyancy. The liquid is taken as incompressible, with the same density at every depth, the balance as perfectly stiff, and the air’s own small buoyancy on the block is not taken off its weight.

The block is flat, twice as wide as it is tall, for a reason. A floating block stays level only if it is wide enough for its density: a cube whose density is between about 21 and 79 percent of the liquid’s is unstable standing level and tips over to float at an angle. A block this flat is stable level at every density, so the level block on screen is where a real one would settle.