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Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

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,[1] the sphericity, , 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:

where is volume of the particle and 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 objectsEdit

The sphericity,  , of an oblate spheroid (similar to the shape of the planet Earth) is:

 

where a and b are the semi-major and semi-minor axes respectively.

DerivationEdit

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,   in terms of the volume of the particle,  

 

therefore

 

hence we define   as:

 

Sphericity of common objectsEdit

Name Picture Volume Surface Area Sphericity
Platonic Solids
tetrahedron        
cube (hexahedron)      

 

octahedron      

 

dodecahedron      

 

icosahedron        
Round Shapes
ideal cone
 
   

 

 

 

 
hemisphere
(half sphere)
     

 

ideal cylinder
 
     

 

ideal torus
 
     

 

sphere      

 

Other Shapes
disdyakis triacontahedron        

See alsoEdit

ReferencesEdit

  1. ^ Wadell, Hakon (1935). "Volume, Shape and Roundness of Quartz Particles". The Journal of Geology. 43 (3): 250–280. Bibcode:1935JG.....43..250W. doi:10.1086/624298.

External linksEdit