= Center of pressure
{disambiguate=fluid mechanics}
{wiki}
The <center of pressure (fluid mechanics)> locates the resultant of a distributed fluid <pressure> load relative to its moment. On a planar surface carrying parallel <pressure> <forces>, the first-moment ratio in <center of pressure> applies. On an airfoil, <pressure> variation and its resultant can change with the flow state, so the point is not a permanently fixed geometric centre. NASA's primary account explicitly distinguishes a moving <center of pressure> from retaining a <force> and pitching moment about an <aerodynamic center>.<Center of Pressure by NASA Glenn Research Center>[Center of Pressure, Angle of Attack, and Aerodynamic Force sections]{ref}
For an original hydrostatic plate example, take a vertical rectangle of width $w$, extending from the free surface to depth $H$ in a constant-density fluid. If the same atmospheric <pressure> acts on its other face, the net gauge <pressure> at depth $y$ is $p(y)=\rho gy$. This hydrostatic relation follows from vertical <force> balance.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[Chapter 40, §40–1, Hydrostatics]{ref} The resultant and its depth are
$$
F=\int_0^H\rho gyw\,dy=\frac12\rho gwH^2,
\qquad
y_P=\frac{\int_0^H y\rho gyw\,dy}{F}=\frac{2H}{3}.
$$
For $\rho=1000\,\mathrm{kg\,m^{-3}}$, $g=9.81\,\mathrm{m\,s^{-2}}$, $w=1\,\mathrm m$ and $H=2\,\mathrm m$, the net <force> is $19.62\,\mathrm{kN}$ at depth $1.333\,\mathrm m$. The area centroid is only one metre below the surface; the increasing <pressure> shifts the load's centroid downward.
A tank demonstration can infer the resultant line from the <force> and hinge moment on a small plate while changing the water depth. The plate's own gravitational moment and the <pressure> on both faces must be included. This tests a specified pressure-load model, not a general claim that the point always lies below every geometric centre.
For aerodynamic net loads, differences of <pressures> on opposite faces can have both signs. The convex-hull property proved for nonnegative contact <pressure> then need not apply, and a small resultant with a finite moment makes its location ill-conditioned. A curved surface or a general <wrench (mechanics)> may also require a residual couple rather than a single <force> that removes every component of <torque>. The resultant convention and moment reference are therefore part of the definition.
Back to article page
