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Archimedes' Law

Forces acting on a body submerged in a fluid

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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

FB =  W

    = V γ

    = V ρ g                                     (1)

where

FB = buoyant force acting on submerged or floating body (N, lbf)

W = weight (gravity force) of displaced liquid (N, lbf)

V = volume of body below surface of liquid (m3, ft3)

γ = ρ g = specific weight of fluid (weight per unit volume) (N/m3, lbf/ft3)

ρ = density of fluid (kg/m3, slugs/ft3)

g = acceleration of gravity (9.81 m/s2, 32.174 ft/s2)

Example - Density of a Body that floats in Water

A floating body is 95% submerged in water with density 1000 kg/m3.

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

FB =  W

or

Vb ρb g = Vw ρw g

where

Vb = volume body (m3)

ρb = density of body (kg/m3)

Vw = volume of water (m3)

ρw  = density of water (kg/m3)

The equation can be transformed to

ρb = Vw ρw / Vb 

Since 95% of the body is submerged

0.95 Vb = Vw

and the density of the body can be calculated as

ρb = 0.95 Vb (1000 kg/m3) / Vb

    = 950 kg/m3

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/ft3. The volume of the brick can be calculated

Vbrick = (3 5/8 in) (2 1/4 in) (8 in)

         = 65.25 in3

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

FB =  ((65.25 in3) / (1728 in/ft3)) (1.940 slugs/ft3) (32.174 ft/s2)  

    = 2.36 lbf

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

WB = (2.36 lbf) 1.75

     = 4.12 lbf

The resulting force acting on the brick can be calculated as

W(WB - FB) = (4.12 lbf) - (2.36 lbf)

   = 1.76 lbf

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