Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape. Defined by Wadell in 1935, the sphericity, $\backslash Psi$, of a particle is the ratio of the surface area of a sphere with the same volume as the given particle to the surface area of the particle:
:$\backslash Psi\; =\; \backslash frac$
where $V\_p$ is volume of the particle and $A\_p$ is the surface area of the particle. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any particle which is not a sphere will have sphericity less than 1.
Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft, is called roundness.

** Ellipsoidal objects **

The sphericity, $\backslash Psi$, of an oblate spheroid (similar to the shape of the planet Earth) is:
:$\backslash Psi\; =\; \backslash frac\; =\; \backslash frac,$
where ''a'' and ''b'' are the semi-major and semi-minor axes respectively.

** Derivation **

Hakon Wadell defined sphericity as the surface area of a
sphere of the same volume as the particle divided by the actual surface area of the particle.
First we need to write surface area of the sphere, $A\_s$ in terms of the volume of the particle, $V\_p$
:$A\_^3\; =\; \backslash left(4\; \backslash pi\; r^2\backslash right)^3\; =\; 4^3\; \backslash pi^3\; r^6\; =\; 4\; \backslash pi\; \backslash left(4^2\; \backslash pi^2\; r^6\backslash right)\; =\; 4\; \backslash pi\; \backslash cdot\; 3^2\; \backslash left(\backslash frac\; r^6\backslash right)\; =\; 36\; \backslash pi\; \backslash left(\backslash frac\; r^3\backslash right)^2\; =\; 36\backslash ,\backslash pi\; V\_^2$
therefore
:$A\_\; =\; \backslash left(36\backslash ,\backslash pi\; V\_^2\backslash right)^\; =\; 36^\; \backslash pi^\; V\_^\; =\; 6^\; \backslash pi^\; V\_^\; =\; \backslash pi^\; \backslash left(6V\_\backslash right)^$
hence we define $\backslash Psi$ as:
:$\backslash Psi\; =\; \backslash frac\; =\; \backslash frac$

** Sphericity of common objects **

** See also **

*Equivalent spherical diameter
*Flattening
*Index of sphericity
*Isoperimetric ratio
*Rounding (sediment)
*Willmore energy

References

** External links **

{{Wiktionary|sphericity

Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

Category:Geometric measurement Category:Spheres Category:Metrology

References

Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

Category:Geometric measurement Category:Spheres Category:Metrology