What is the buoyancy force?

How it works

Archimedes' principle: the buoyant force on an object equals the weight of the fluid it pushes aside. The fluid pushed aside has the volume V of the object under the surface, so its mass is ρ × V and its weight is

F_B = ρ × V × g,

with ρ the fluid density in kg/m³, V in m³ and g in m/s² (9.80665 m/s² is standard gravity). The page shows F_B in N and in lbf (F_B ÷ 4.4482216152605), the mass of fluid pushed aside, ρ × V, in kg, and the density used.

Fluids (kg/m³, OpenStax Tables 14.1 and 14.2): fresh water at 4 °C 1,000; at 20 °C 998.2; at 100 °C 958.4; seawater at 0 °C 1,030; mercury 13,600; glycerin 1,260; olive oil 920; benzene 879; ethyl alcohol 806; gasoline 680 (the liquids of Table 14.1 are at 0 °C); air at 0 °C and 101.3 kPa 1.29. Or type your own density.

With the object's mass m (optional), V is the object's whole volume, held fully under the surface:

• Weight W = m × g, in N.
• Apparent weight W − F_B, in N: what a scale under the surface reads. A value below 0 means the fluid pushes the object up.
• Object density m ÷ V, in kg/m³.
• Floats or sinks, comparing m ÷ V with ρ exactly: "Sinks: it is denser than the fluid" when m ÷ V > ρ, "Floats: it is less dense than the fluid" when m ÷ V < ρ, and "Neither: it has the same density as the fluid" when they are equal.
• When it floats (or neither), the share of its volume under the surface once it floats freely: 100 × (m ÷ V) ÷ ρ, in percent. A floating object sinks until the fluid it pushes aside weighs as much as the object.

Units (exact): 1 mL = 1 cm³ = 10⁻⁶ m³; 1 L = 0.001 m³; 1 in³ = 0.000016387064 m³; 1 ft³ = 0.028316846592 m³; 1 gal = 0.003785411784 m³; 1 g/cm³ = 1,000 kg/m³; 1 lb/ft³ = 0.45359237 ÷ 0.028316846592 kg/m³ (16.0185 kg/m³); 1 g = 0.001 kg; 1 lb = 0.45359237 kg; 1 oz = 0.028349523125 kg.

Rules

• Volume from 10⁻¹² m³ to 10⁹ m³; mass from 10⁻¹² kg to 10¹² kg; a typed fluid density from 10⁻⁶ to 10⁶ kg/m³; g from 10⁻⁶ to 10⁶ m/s². Limits are in SI units and include their ends; a value typed in another unit is converted first.
• The fluid is at rest and has one density throughout.

Output format

Every number shows to 6 significant figures, rounded half up, with trailing zeros dropped, thousands commas, the true minus sign (−) for a negative apparent weight, and from 10¹⁵ up or below 10⁻⁶ as a × 10ⁿ. Typed values are read back as the exact decimals you typed and every unit size and table density is exact, so each result is an exact fraction rounded once, and floats or sinks compares exact values.

The answer sentence reads The buoyant force is F N (F′ lbf)., with F and F′ written as their outputs are.

Worked examples by hand

10 litres under fresh water. ρ = 1,000 kg/m³, V = 0.01 m³, g = 9.80665 m/s²: F_B = 1,000 × 0.01 × 9.80665 = 98.0665 N, which is 98.0665 ÷ 4.4482216152605 = 22.0462 lbf: the weight of the 10 kg of water pushed aside.

One cubic foot of fresh water. F_B = 1,000 × 0.028316846592 × 9.80665 = 277.693 N = 62.428 lbf.

Seawater. 1 m³ under seawater (1,030 kg/m³) with g = 9.8 m/s²: F_B = 1,030 × 1 × 9.8 = 10,094 N.

A floater (OpenStax Example 14.4). An object of 97 kg and 0.1 m³: its density is 97 ÷ 0.1 = 970 kg/m³, less than fresh water's 1,000, so it floats, with 970 ÷ 1,000 = 97% under the surface, the 97.0% of the example. Held fully under with g = 9.8: W − F_B = 950.6 − 980 = −29.4 N, so the water pushes it up.

A sinker. 5 kg of 2 L (0.002 m³) in fresh water: 2,500 kg/m³, so it sinks. W = 5 × 9.80665 = 49.03325 N, F_B = 1,000 × 0.002 × 9.80665 = 19.6133 N, and it weighs 29.41995 N under water.

Air. 1 m³ of air at 0 °C (1.29 kg/m³): F_B = 1.29 × 1 × 9.80665 = 12.6506 N, the lift on any object of that volume before its own weight.