Parallel axis theorem: Difference between revisions

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 axis z is given by:

${\displaystyle I_{z}=I_{cm}+mr^{2},\,}$

where:

${\displaystyle I_{cm}\!}$ is the moment of inertia of the object about an axis passing through its centre of mass;

${\displaystyle m\!}$ is the object's mass;

${\displaystyle r\!}$ is the perpendicular distance between the axis of rotation and the axis that would pass through the centre of mass.

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) for a plane region D:

${\displaystyle I_{z}=I_{x}+Ar^{2},\,}$

where:

${\displaystyle I_{z}\!}$ is the area moment of inertia of D relative to the parallel axis;

${\displaystyle I_{x}\!}$ is the area moment of inertia of D relative to its centroid;

${\displaystyle A\!}$ is the area of the plane region D;

${\displaystyle r\!}$ is the distance from the new axis z to the centroid of the plane region D.

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

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:

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

The moment of inertia relative to the new axis, perpendicular distance r along the x-axis from the centre of mass, is:

${\displaystyle I_{z}=\int {((x-r)^{2}+y^{2})}dm}$

If we expand the brackets, we get:

${\displaystyle I_{z}=\int {(x^{2}+y^{2})}dm+r^{2}\int dm-2r\int {x}dm}$

The first term is I_cm, the second term becomes mr², and the final term is zero since the origin is at the centre of mass. So, this expression becomes:

${\displaystyle I_{z}=I_{cm}+mr^{2}\,}$

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 centre of mass I_ij when the pivot point is a displacement a from the centre of mass:

${\displaystyle J_{ij}=I_{ij}+m(\|{\boldsymbol {a}}\|^{2}\delta _{ij}-a_{i}a_{j})\!}$

where

${\displaystyle {\boldsymbol {a}}=a_{1}{\boldsymbol {\hat {x}}}+a_{2}{\boldsymbol {\hat {y}}}+a_{3}{\boldsymbol {\hat {z}}}\!}$

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

${\displaystyle \delta _{ij}\!}$

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.

• Perpendicular axis theorem
• Stretch rule
• Jakob Steiner
• Christiaan Huygens

• Parallel axis theorem