Parallel axis theorem

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

Mass moment of inertia

Suppose a body of mass m is rotated about an axis z passing through the body's center of mass. The body has a moment of inertia Icm with respect to this axis.

The parallel axis theorem states that if the body is made to rotate instead about a new axis z′, which is parallel to the first axis and displaced from it by a distance d, then the moment of inertia I with respect to the new axis is related to Icm by

I

=

I

c

m

+

m

d

2

.

{\displaystyle I=I_{\mathrm {cm} }+md^{2}.}

Explicitly, d is the perpendicular distance between the axes z and z′.

The parallel axis theorem can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.

Derivation

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 center of mass lies at the origin. The moment of inertia relative to the z-axis is then

I

c

m

=

∫

(

x

2

+

y

2

)

d

m

.

{\displaystyle I_{\mathrm {cm} }=\int (x^{2}+y^{2})\,dm.}

The moment of inertia relative to the axis z′, which is at a distance D from the center of mass along the x-axis, is

I

=

∫

[

(

x

−

D

)

2

+

y

2

]

d

m

.

{\displaystyle I=\int \left[(x-D)^{2}+y^{2}\right]\,dm.}

Expanding the brackets yields

I

=

∫

(

x

2

+

y

2

)

d

m

+

D

2

∫

d

m

−

2

D

∫

x

d

m

.