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Moment of Inertia


The moment of inertia with respect to a given axis of a solid body with density rho(r) is defined by the volume integral

 I=intrho(r)r__|_^2dV,
(1)

where r__|_ is the perpendicular distance from the axis of rotation. This can be broken into components as

 I_(jk)=sum_(i)m_i(r_i^2delta_(jk)-x_(i,j)x_(i,k))
(2)

for a discrete distribution of mass, where r is the distance to a point (not the perpendicular distance) and delta_(jk) is the Kronecker delta, or

 I_(jk)=int_Vrho(r)(r^2delta_(jk)-x_jx_k)dV
(3)

for a continuous mass distribution. Depending on the context, I may be viewed either as a tensor or a matrix. Expanding (3) in terms of Cartesian axes gives the equation

 I=int_Vrho(x,y,z)[y^2+z^2 -xy -xz; -xy z^2+x^2 -yz; -xz -yz x^2+y^2]dxdydz.
(4)

The moment of inertia of a region can be computed in the Wolfram Language using MomentOfInertia[reg].

The moment of inertia tensor I is symmetric, and is related to the angular momentum vector L by

 L=Iomega,
(5)

where omega is the angular velocity vector.

The principal moments of inertia are given by the entries in the diagonalized moment of inertia matrix, and are denoted (for a solid) A, B, and C in order of decreasing magnitude. In the principal axes frame, the moments are also sometimes denoted I_(xx), I_(yy), and I_(zz). The principal axes of a rotating body are defined by finding values of I such that

 L=[L_x; L_y; L_z]=[I_(11) I_(12) I_(13); I_(21) I_(22) I_(23); I_(31) I_(32) I_(33)][omega_x; omega_y; omega_z]=I[omega_x; omega_y; omega_z]
(6)
 [I_(11)-I I_(12) I_(13); I_(21) I_(22)-I I_(23); I_(31) I_(32) I_(33)-I][omega_x; omega_y; omega_z]=[0; 0; 0],
(7)

which is an eigenvalue problem.

The following table summarizes the moments of inertia of some common solids around some of their principal axes.

solid/axismoment of inertia
cylinder about symmetry axis1/2MR^2
cylinder about central diameter1/(12)Mh^2+1/4MR^2
ellipsoid about principal axis1/5M(b^2+c^2)
elliptical slab about major axis1/6M(3b^2+4h^2)
elliptical slab about vertical1/2M(a^2+b^2)
cuboid about major axis1/3(b^2+c^2)
ring about perpendicular axisMR^2
ring about diameter1/2MR^2
rod about end1/3Mh^2
rod about center1/(12)Mh^2
sphere about diameter2/5MR^2
spherical shell2/3MR^2
torus about diameter1/8(5a^2+4c^2)M
torus about symmetry axis(3/4a^2+c^2)M

See also

Area Moment of Inertia, Radius of Gyration

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References

Dobrovolskis, A. R. "Inertia of Any Polyhedron." Icarus 124, 698-704, 1996.Lawlor, O. "Boundary Integration and the Rotational Inertia Matrix." CS 482 Lecture. https://www.cs.uaf.edu/2015/spring/cs482/lecture/02_20_boundary.html.

Referenced on Wolfram|Alpha

Moment of Inertia

Cite this as:

Weisstein, Eric W. "Moment of Inertia." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MomentofInertia.html

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