Hydrostatics, or [[Fluid|fluid]] [[Static Systems|statics]], is the study of fluids at rest. Important applications of hydrostatics include buoyancy and hydrostatic pressure. A static fluid in Earth’s atmosphere will only have two [[Force|forces]] acting upon it: - [[Gravity]]: Due to Earth’s gravitational field. - [[Fluid Pressure]]: Due to air and fluid above a particular fluid particle at a given point. The force due to [[Pressure|pressure]] is: $F=-\iiint_{V\llap{--}}(\nabla p)dV\llap{--}$ >[!proof]- $F=-\iint_{A_x}pdA_x\hat{i} - \iint_{A_y}pdA_y\hat{j} − \iint_{A_z}pdA_z\hat{k}$ $F=-\iint_{A_x}pdydz\hat{i}-\iint_{A_y}pdxdz\hat{j}-\iint_{A_z}pdxdy\hat{k}$ $F=-\iint_{A_x}\dfrac{\partial p}{\partial x}dxdydz\hat{i}-\iint_{A_y}\dfrac{\partial p}{\partial y}dxdydz\hat{j}-\iint_{A_z}\dfrac{\partial p}{\partial z}dxdydz\hat{k}$ $F=-\iiint_{V\llap{--}}(\nabla p)dV\llap{--}$ Due to gravity, fluid pressure varies with fluid depth. The effect of pressure can therefore be expressed as the follows: $\nabla p=\rho g$ Or more simply: $\dfrac{dp}{dz}=\rho g$ From which we can conclude the familiar formula for pressure at a given depth in a fluid: $p=p_0+\rho gz$ For the simplified 2D case of a submerged object where $y$ is aligned with the force of gravity instead of $z$, the following equation applies for the force due to fluid pressure: $F=\iint_A−(\rho gy)dA$ The pressure force always acts [[Normal|normal]] to a surface, therefore we can determine the direction of the force independently from the magnitude of the force. To simplify analysis we define the point, $(x_c,y_c)$, where the sum of all forces and [[Moment|moments]] due to fluid pressure can be described by a single equivalent force. This point is termed the **centre of pressure**. $x_c=\dfrac{1}{F}\iint_{A_x}PdA$ $y_c=\dfrac{1}{F}\iint_{A_y}PdA$