- Parallel axis theorem
In

physics , the**parallel axis theorem**can be used to determine themoment of inertia of arigid body about any axis, given the moment of inertia of the object about the parallel axis through the object'scenter of mass and theperpendicular distance between the axes.Let:

"I"_{"CM"}denote the moment of inertia of the object about the centre of mass,

"M" the object's mass and "d" the perpendicular distance between the two axes.Then the moment of inertia about the new axis "z" is given by:

:$I\_z\; =\; I\_\{cm\}\; +\; Md^2.,$

This rule can be applied with the

stretch rule andperpendicular axis theorem to find moments of inertia for a variety of shapes.The parallel axes rule also applies to the

second moment of area (area moment of inertia);:$I\_z\; =\; I\_x\; +\; Ad^2.,$where:

"I_{z}" is the area moment of inertia through the parallel axis,

"I_{x}" is the area moment of inertia through the centre of mass of thearea ,

"A" is the surface of the area, and

"d" is the distance from the new axis "z" to the centre of gravity of the area.The parallel axis theorem is one of several theorems referred to as

**Steiner's theorem**, afterJakob Steiner .**In classical mechanics**In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor

**J**from an inertia tensor about a center of mass_{ij}**I**when the pivot point is a displacement_{ij}**a**from the center of mass::$J\_\{ij\}=I\_\{ij\}\; +\; M(a^2\; delta\_\{ij\}-a\_ia\_j)$

where

:$\backslash boldsymbol\{a\}=a\_1\backslash boldsymbol\{hat\{x+a\_2\backslash boldsymbol\{hat\{y+a\_3\backslash boldsymbol\{hat\{z$

is the displacement vector from the center of mass to the new axis, and

:$delta\_\{ij\}$

is the

Kronecker delta .We can see that, for diagonal elements (when "i" = "j"), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.

**ee also***

Perpendicular axis theorem

*Stretch rule **References**[

*http://scienceworld.wolfram.com/physics/ParallelAxisTheorem.html Parallel axis theorem*]

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