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According to the law of floatation when a body of surface area S is floating the liquid with density $\rho$ then

Weight of the body =weight of the displaced liquid. But why don't we count the force due to the atmospheric pressure here making it

Weight of the body + $P_0S$=weight of the displaced liquid.

I have started learning fluid dynamics hence please don't find if its a very foolish question.

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  • $\begingroup$ The atmospheric pressure also increases the pressure of the liquid below the surface, and, if you include the contributions of atmospheric pressure above and below the surface, they cancel out. $\endgroup$
    – Chet Miller
    Commented Nov 2, 2021 at 1:01

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Because if the pressure at the surface if the liquid is $P_{\rm atmosphere}$ and the pressure below the surface of the liquid is $P_{\rm atmosphere + liquid}$ then for the upthrust you are interested in the difference in pressure due to the liquid at the bottom and top of the submerged part of the floating body.$\Delta P_{\rm liquid}$ which is $P_{\rm atmosphere + liquid} - P_{\rm atmosphere} = \Delta P_{\rm liquid}$.

There is an upthrust due to the atmosphere because the pressure due to the atmosphere at the surface of the liquid is larger than the pressure due to the atmosphere at the top of the floating object but as this pressure difference is much smaller than the pressure difference due to the liquid it is usually neglected.

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  • $\begingroup$ I still don't get it. There is surely a force on the floating object due to atmosphere. So how can we neglect it when we make the FBD of the floating object? $\endgroup$
    – user118752
    Commented Oct 12, 2016 at 9:03
  • $\begingroup$ & there is also a force on the surface of the liquid due to the atmosphere. That force due to the atmosphere on the surface of the liquid is transmitted by the liquid in the same way as the force due to atmosphere on the top of the object is transmitted by the object. At the level of the bottom of the object the liquid must exert an upward force to balance the force due to the weight of liquid above it & the force due to the atmosphere. The liquid is exerting an upward force equal to that due to the atmosphere & the object is exerting a downward force equal to that exerted by the atmosphere. $\endgroup$
    – Farcher
    Commented Oct 12, 2016 at 9:25

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