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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, \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: :\Psi = \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, \Psi , of an oblate spheroid (similar to the shape of the planet Earth) is: :\Psi = \frac = \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 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac r^6\right) = 36 \pi \left(\frac r^3\right)^2 = 36\,\pi V_^2 therefore :A_ = \left(36\,\pi V_^2\right)^ = 36^ \pi^ V_^ = 6^ \pi^ V_^ = \pi^ \left(6V_\right)^ hence we define \Psi as: : \Psi = \frac = \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