Parallel axis theorem

In physics, the parallel axis theorem can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object's center of mass and the perpendicular 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 and perpendicular 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 the area,
"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, after Jakob 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_ij from an inertia tensor about a center of mass I_ij when the pivot point is a displacement a from the center of mass:

: $J_{ij}=I_{ij} + M(a^2 delta_{ij}-a_ia_j)$

where

: $oldsymbol{a}=a_1oldsymbol{hat{x+a_2oldsymbol{hat{y+a_3oldsymbol{hat{z$

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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