Sphericity is a measure of how closely the shape of a

physical object In natural language and physical science, a physical object or material object (or simply an object or body) is a contiguous collection of matter, within a defined boundary (or surface), that exists in space and time. Usually contrasted with ...

resembles that of a perfect

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. For example, the sphericity of the balls inside a

ball bearing A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...

determines the

quality Quality may refer to: Concepts *Quality (business), the ''non-inferiority'' or ''superiority'' of something *Quality (philosophy), an attribute or a property *Quality (physics), in response theory *Energy quality, used in various science discipli ...

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. Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a

cylindrical A cylinder () has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder may ...

object such as a shaft, is called ''roundness''.

Definition

Defined by Wadell in 1935, the sphericity,

\Psi

, of an object is the ratio of the

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

of a sphere with the same volume to the object's surface area: :

\Psi = \frac

where

V_p

is volume of the object and

A_p

is the surface area. The sphericity of a sphere is unity by definition and, by the

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' ...

, any shape which is not a sphere will have sphericity less than 1.

Ellipsoidal objects

The sphericity,

\Psi

, of an

oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...

(similar to the shape of the planet

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

) 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 object divided by the actual surface area of the object. First we need to write surface area of the sphere,

A_s

in terms of the volume of the object being measured,

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

References

External links