According to Newton’s second law: $ \Sigma\vec{F}_S=0 $ Applying this to a fluid results in the hydrostatic equation: $ -\nabla p+\rho\vec{g}=0 $ If we choose a Cartesian coordinate system with the $z$ axis oriented upward. this reduces to $ \frac{dp}{dz}=-\rho g $ To find pressure variation, boundary conditions are applied and the equation is integrated: $ \int_{p_0}^pdp=-\int_{z_0}^z\rho g\,dz $ $ p-p_0=-\rho g(z-z_0)=\rho g(z_0-z) $ Measuring height $h$ downward as $h=z_0-z$: $ \Delta p=\rho gh $ ## Hydrostatic Forces on General Submerged Surfaces if we know the pressure distribution in a fluid, we can determine the net pressure acting on any surface, given by the net pressure force vector: $ \vec{F}_p=-\iint_Ap\hat{n}dA $ The shape of the surface must be known to compute this. Moment vector: $ \vec{M}_p=-\vec{r}\times\iint_Ap\hat{n}dA $ [[Buoyancy]] [[Barometer]] [[Manometer]]