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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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