Buoyant force

A stone hanging from a force meter gives a lower reading when immersed in water. Its mass has not disappeared. Which forces have changed, and why does water push the stone upwards at all?

Imagine a rectangular block immersed in water. Water pushes down on its top face and up on its bottom face. The bottom is deeper, so its pressure and force are greater for equal face areas. Sideways forces at equal depths cancel. Adding all the forces leaves an upward force: buoyant force, FА. Water does not act only from below; the lift comes from pressure acting over the entire immersed surface.

How can we avoid calculating pressure on every face of a complicated object? Imagine replacing it with the same liquid, filling the space it occupied. In a stationary medium, this portion would remain in equilibrium: external pressure forces would balance its gravitational force. The surrounding liquid acts on the same boundary in the same way when the object replaces that portion. Therefore, buoyant force equals the weight of the displaced liquid. This is Archimedes' principle.

We need the volume from which the object excludes the medium, Vпогр, measured in m³. For a solid or sealed object fully immersed, it is the external volume; for a floating object, it is the underwater part. A cavity freely filled by water does not itself displace water. Imagine an open cup and a sealed box of the same external size: once water enters the cup, their displaced volumes can differ.

The mass of the displaced uniform medium is ρсредыVпогр, and its weight is this product multiplied by g. Thus FА = ρсреды·g·Vпогр. Doubling the immersed volume doubles the force; a denser medium pushes more strongly on the same volume. Using kg/m³, N/kg and m³ gives N. The object's own density does not appear in this formula: two fully immersed blocks of equal volume experience equal buoyant forces, although their gravitational forces may differ.

Lower a rigid, fully immersed block deeper into a uniform incompressible liquid. Pressure above and below increases equally, so the difference stays the same; Vпогр and FА also remain unchanged. Here the medium is stationary and surrounds the object, including underneath. If the object is pressed tightly against the bottom so water cannot reach beneath it, the set of pressure forces changes: the usual formula needs a separate analysis. A compressible object may also change its displaced volume.

Hang an object from a force meter and hold it stationary first in air, then in water, without touching the sides or bottom. In water, both the suspension and buoyancy act upwards, so the suspension needs less force for equilibrium. Neglecting air buoyancy gives FА = Pвозд − Pжидк, where P denotes force-meter readings. The difference measures the support provided by water. The gravitational force on the object changes negligibly when immersed.

The same reasoning applies to a gas: its pressure in a gravitational field increases downwards, and the difference in pressure forces produces an upward buoyant force. Use the gas density in the formula, not water density. For a small dense object, air buoyancy is weak because air has low density; a large balloon displaces a substantial mass of air. To predict lift, we must also compare buoyancy with the total gravitational force on the balloon. That comes next.

A block is fully immersed in water; V = 0,002 м³, ρводы = 1000 кг/м³, g = 10 Н/кг. Find FА.

We need the volume of displaced water. For full immersion, it equals the block's volume.

FА = 1000·10·0,002 = 20 Н.

This is the weight of two kilograms of displaced water. The block's own mass is not needed for this calculation.

The buoyant force is 20 N upwards.

Buoyancy comes from the difference in pressure forces, not from gravity disappearing.

FА equals the weight of the displaced medium: use its density and the volume actually displaced.

Greater depth alone does not increase buoyancy on a rigid object in a uniform liquid; how the medium contacts the object matters.

To predict floating, compare the buoyant force we found with the gravitational force on the object.