# Archimedes' Law

Archimedes' principle states that:

* "If a solid body floats or is submerged in a liquid - the liquid exerts an upward thrust force - a buoyant force - on the body equal to the gravitational force on the liquid displaced by the body." *

The buoyant force can be expressed as

* F _{ B } = W *

* = V γ *

* = V ρ g (1) *

* where *

* F _{ B } = buoyant force acting on submerged or floating body (N, lb_{f} ) *

* W = weight (gravity force) of displaced liquid (N, lb _{f} ) *

* V = volume of body below surface of liquid (m ^{3}, ft^{3} ) *

* γ = ρ g = specific weight of fluid (weight per unit volume) (N/m ^{3}, lb_{f} /ft^{3} ) *

* ρ = density of fluid (kg/m ^{3}, slugs/ft^{3} ) *

* g = acceleration of gravity (9.81 m/s ^{2}, 32.174 ft/s^{2}) *

### Example - Density of a Body that floats in Water

A floating body is * 95% * submerged in water with density * 1000 kg/m *^{3}.

For a floating body the buoyant force is equal to the weight of the water displaced by the body.

* F _{ B } = W *

* or *

* V _{ b } ρ _{ b } g = V _{ w } ρ _{ w } g *

* where *

* V _{ b } = volume body (m^{3} ) *

* ρ _{ b } = density of body ( kg/m^{3} ) *

* V _{ w } = volume of water (m^{3} ) *

* ρ _{ w } = density of water ( kg/m^{3} ) *

The equation can be transformed to

* ρ _{ b } = V _{ w } ρ _{ w } / V _{ b } *

Since 95% of the body is submerged

* 0.95 V _{ b } = V _{ w } *

and the density of the body can be calculated as

* ρ _{ b } = 0.95 V _{ b } (1000 kg/m^{3} ) / V _{ b } *

* = 950 kg/m ^{3} *

### Example - Buoyant force acting on a Brick submerged in Water

A standard brick with actual size * 3 5/8 x 2 1/4 x 8 (inches) * is submerged in water with density * 1.940 slugs /ft ^{3} * . The volume of the brick can be calculated

* V _{ brick } = (3 5/8 in) (2 1/4 in) (8 in) *

* = 65.25 in ^{3} *

The buoyant force acting on the brick is equal to the weight of the water displaced by the brick and can be calculated as

* F _{ B } = (( 65.25 in^{3} ) / (1728 in/ft^{3} )) ( 1.940 slugs/ft^{3} ) ( 32.174 ft/s^{2}) *

* = 2.36 lb _{f} *

The weight or the gravity force acting on the brick - common red brick has specific gravity 1.75 - can be calculated to

* W _{ B } = (2.36 lb_{f} ) 1.75 *

* = 4.12 lb _{f} *

The resulting force acting on the brick can be calculated as

* W _{ (WB - FB) } = (4.12 lb_{f} ) - (2.36 lb_{f} ) *

* = 1.76 lb _{f} *

## Related Topics

### • Fluid Mechanics

The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.

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