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Consider an object being immersed through a height h in a fluid. The pressure at the surface is smaller than the pressure at the bottom of the object. This difference in pressure, Dp, is given by:
Dp = hdg
where d is the density of the fluid
and g is the acceleration of free- fall
Now, Dp can be expanded as the ratio of force to area. So:
F/A = hdg
where F is the force exerted on the body
and A is the bottom area of the object
=> F = Ahdg
=> F = Vdg
since Ah = volume V
=> F = mfg
where mf is the mass of fluid in the volume V ( mf = dV from the definition of density)
=> F = Weight of fluid displaced (shown)
The force F is the upthurst on the body.
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