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Cause of Upthrust

Cause of Upthrust

Consider a cylindrical solid of cross-section area A which is totally immersed in a fluid of density ρ as shown.

The pressure due to liquid column is usually given by P=ρgh.

Pressure at the top of the solid will be given by, PT = h1ρg.

Where h1 is the height of the liquid column above the top of the object.

Pressure at the lower end of the object will be given by

Pb=h2ρg where h2 is the height of the liquid above the lover surface of the cylinder .

The pressure at the top of the cylinder will provide downward force exerted by the liquid up on the object.

From the pressure laws, F=pressure P x Area A.

i.e F=PA.

Taking the area of the cylinder at the top, the force from the liquid acting on that surface is Given by F=PT x A=h1ρgA.

Similarly, pressure at the bottom is given as F=PB x A=h2ρgA.

The total resultant upwardward force F is this given as

F=F2-F1

Hence F=h2ρgA-h1ρgA

Factoring out the common factors: F=ρgA (h2-h1)

Let h be the difference between liquid column on top and the one at bottom h2 such that h=h2-h1

Hence F=ρgAh

But Volume is always given by V=Ah

The resultant force F is the upthrust force U and will thus be expressed as.

F=U=Aρpg=pgV

where V is the volume of the liquid displaced.

Mass of the liquid is usually given by density x volume. Hence mass m of liquid displaced will be given by m=Ahρ

Weight is usually given as Weight W=mg

Hence weight of liquid displaced will be W=U=Ahρg which represents the upthrust force we calculated earlier. This confirms the archimedes principle that upthrust force is equal to the weight of the fluid it displaces.

From our mathematical arguments, it should be easy to see that Magnitude of the upthrust force is equal a function of volume of the object and density of the liquid considering that gravitational pull g is a constant.

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