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Parallel axis theorem 1

Parallel axis theorem

In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of

area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel

axis through the object's centre of mass and the perpendicular distance (r) between the axes.

The moment of inertia about the new axisz is given by:

where:

is the moment of inertia of the object about an axis passing through its centre of mass;

is the object's mass;

is the perpendicular distance between the two axes.

This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety

of shapes.

Parallel axes rule for area moment of inertia

The parallel axes rule also applies to the second moment of area (areamoment of inertia) for a plane regionD:

where:

is the area moment of inertia ofD relative to the parallel axis;

is the area moment of inertia ofD relative to its centroid;

is the area of the plane regionD;

is the distance from the new axisz to the centroid of the plane regionD.

Note: The centroid ofD coincides with the centre of gravity (CG) of a physical plate with the same shape that has

uniform density.

Proof

We may assume, without loss of generality, that in a Cartesian coordinate system the perpendicular distance between

the axes lies along the x-axis and that the centre of mass lies at the origin. The moment of inertia relative to the

z-axis, passing through the centre of mass, is:

The moment of inertia relative to the new axis, perpendicular distancer along thex-axis from the centre of mass, is:

If we expand the brackets, we get:

http://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_systemhttp://en.wikipedia.org/w/index.php?title=Centre_of_gravityhttp://en.wikipedia.org/w/index.php?title=Centroidhttp://en.wikipedia.org/w/index.php?title=Centroidhttp://en.wikipedia.org/w/index.php?title=Centroidhttp://en.wikipedia.org/w/index.php?title=Second_moment_of_areahttp://en.wikipedia.org/w/index.php?title=File%3AParallelaxes-1.pnghttp://en.wikipedia.org/w/index.php?title=Perpendicular_axis_theoremhttp://en.wikipedia.org/w/index.php?title=Stretch_rulehttp://en.wikipedia.org/w/index.php?title=Distancehttp://en.wikipedia.org/w/index.php?title=Perpendicularhttp://en.wikipedia.org/w/index.php?title=Centre_of_masshttp://en.wikipedia.org/w/index.php?title=Parallel_%28geometry%29http://en.wikipedia.org/w/index.php?title=Rigid_bodyhttp://en.wikipedia.org/w/index.php?title=Mass_moment_of_inertiahttp://en.wikipedia.org/w/index.php?title=Second_moment_of_areahttp://en.wikipedia.org/w/index.php?title=Second_moment_of_areahttp://en.wikipedia.org/w/index.php?title=Physics

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Parallel axis theorem 2

The first term isIcm

, the second term becomesmr2

, and the final term is zero since the origin is at the centre of mass.

So, this expression becomes:

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 Jij

from an inertia tensor about a centre of mass Iij

when the pivot point is a

displacementa from the centre of mass:

where

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

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.

References

Parallel axis theorem[1]

References

[1] http://scienceworld.wolfram.com/physics/ParallelAxisTheorem.html

http://scienceworld.wolfram.com/physics/ParallelAxisTheorem.htmlhttp://scienceworld.wolfram.com/physics/ParallelAxisTheorem.htmlhttp://en.wikipedia.org/w/index.php?title=Kronecker_deltahttp://en.wikipedia.org/w/index.php?title=Moment_of_inertia%23Moment_of_inertia_tensor

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Article Sources and Contributors 3

Article Sources and ContributorsParallel axis theorem Source: http://en.wikipedia.org/w/index.php?oldid=429319880 Contributors: ABCD, Ablewisuk, Acidwillburnyou, CapitalR, Charles Matthews, Currir55, Dbfirs,

Deeptrivia, Dger, Giftlite, Gombang, Hean.excogitate, Ideal gas equation, Jogers, Keenan Pepper, Loodog, MagneticFlux, MarcusMaximus, Melchoir, Mercury, Michael Hardy, Mykar15,

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anonymous edits

Image Sources, Licenses and ContributorsImage:Parallelaxes-1.png Source: http://en.wikipedia.org/w/index.php?title=File:Parallelaxes-1.png License: Creative Commons Attribution-Sharealike 3.0 Contributors: D. Gordon E.

Robertson

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