Moment of inertia

This article is about the moment of inertia of a rotating object. For the moment of inertia dealing with bending of a plane, see second moment of area.

Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft²) quantifies the rotational inertia of a rigid body, i.e. its inertia with respect to rotational motion, in a manner somewhat analogous to how mass quantifies the inertia of a body with respect to translational motion. The symbols $I$ and sometimes $J$ are usually used to refer to the moment of inertia.

See also moment (physics)

Overview

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. For example, consider two discs of the same mass, one large and one small in radius. Assuming that there is uniform thickness and mass distribution, the larger radius disc requires more effort to accelerate it (i.e. change its angular motion) because its mass is effectively distributed further from its axis of rotation. Conversely, the smaller radius disc takes less effort to accelerate it because its mass is distributed closer to its axis of rotation. Quantitatively, the larger disc has a larger moment of inertia, whereas the smaller disc has a smaller moment of inertia.

The moment of inertia has two forms, a scalar form $I$ (used when the axis of rotation $\mathbf {\hat {n}}$ is known) and a more general tensor form $\mathbf {I}$ that does not require knowing the axis of rotation. The scalar moment of inertia $I$ is often called simply the "moment of inertia".

The moment of inertia can also be called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol $I$. The easiest way to differentiate these quantities is through their units.

In addition, the moment of inertia should not be confused with the polar moment of inertia, which is a measure of an object's ability to resist torsion (twisting).

Definition

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

$I\ {\stackrel {\mathrm {def} }{=}}\ mr^{2}\,\!$

where

m is its mass,

and r is its perpendicular distance from the axis of rotation.

The moment of inertia is additive so, for a rigid body consisting of $N$ point masses $m_{i}$ with distances $r_{i}$ to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia

$I\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=1}^{N}{m_{i}r_{i}^{2}}\,\!$

to a solid body described by a continuous mass-density function $\mathbf {\rho }$, the moment of inertia for rotating about a known axis can be calculated by integrating the moments of the point masses relative to the rotation axis

$I\ {\stackrel {\mathrm {def} }{=}}\ \int _{V}r^{2}(m)\,dm=\iiint _{V}r^{2}(v)\,\rho (v)\,dv=\iiint _{V}r^{2}(x,y,z)\,\rho (x,y,z)\,dx\,dy\,dz\!$

where

V is the volume region of the object,

r is the distance from the axis of rotation,

m is mass,

v is volume,

ρ is the pointwise density function of the object,

and x, y, z are the Cartesian coordinates.

The moment of inertia for non-point objects can also be found or approximated as the product of three terms:

$I=k\cdot M\cdot {R}^{2}\,\!$

where

$k$ is the inertial constant,

$M$ is the mass, and

$R$ is the radius of the object from the center of mass.

Inertial constants are used to account for the differences in the placement of the mass from the center of rotation. Placing all the mass on the outside of the disk would provide for the biggest inertial constant. For example:

• $k=1\$, thin ring or thin-walled cylinder around its center,
• $k=2/5\$, solid sphere around its center
• $k=1/12\$, thin rod around its center.

Parallel axis theorem

If the moment of inertia has been calculated for rotations about the center of mass (usually the centroid) of a rigid body, we can conveniently recalculate the moment of inertia for all parallel rotation axes as well, without having to resort to the formal definition. If the axis of rotation is displaced by a distance $R$ from the centroid axis of rotation (e.g., spinning a disc about a point on its periphery, rather than through its center), the new moment of inertia equals:

$I_{\mathrm {displaced} }=I_{\mathrm {centroid} }+MR^{2}\,\!$

where

$M$ is the total mass of the rigid body, and

R is the distance of the axis of rotation from the centroid axis of rotation (as described above).

This theorem is also known as parallel axes rule or Huygens-Steiner's theorem.

Kinetic energy

The kinetic energy of a system can be expressed in terms of its moment of inertia. For a system with $N$ point masses $m_{i}$ moving with speeds $v_{i}$, the kinetic energy $T$ always equals

$T=\sum _{i=1}^{N}{\frac {1}{2}}m_{i}v_{i}^{2}\,\!$

For a rigid body rotating with angular speed $\omega$, the speeds can be written

$v_{i}=\omega r_{i}\,\!$(omega is in rad/sec)

where again $r_{i}$ is the shortest distance from the point mass to the rotation axis. Therefore, the kinetic energy can be written

$T=\sum _{i=1}^{N}{\frac {1}{2}}m_{i}\omega ^{2}r_{i}^{2}={\frac {1}{2}}I\omega ^{2}\,\!$

The final formula $T={\frac {1}{2}}I\omega ^{2}\,\!$ also holds for a continuous distribution of mass.

Angular momentum and torque

Similarly, the angular momentum $\mathbf {L}$ for a system of particles with linear momenta $p_{i}$ and distances $r_{i}$ from the origin is defined

$\mathbf {L} =\sum _{i=1}^{N}\mathbf {r} _{i}\times \mathbf {p} _{i}=\sum _{i=1}^{N}m_{i}\mathbf {r} _{i}\times \mathbf {v} _{i}$

For a rigid body rotating with angular velocity $\omega$ about the rotation axis $\mathbf {\hat {n}}$ (a unit vector), the velocity vector $\mathbf {v} _{i}$ may be written as a vector cross product

$\mathbf {v} _{i}=\omega \mathbf {\hat {n}} \times \mathbf {r} _{i}\ {\stackrel {\mathrm {def} }{=}}\ {\boldsymbol {\omega }}\times \mathbf {r} _{i}$

where

angular velocity vector ${\boldsymbol {\omega }}\ {\stackrel {\mathrm {def} }{=}}\ \omega \mathbf {\hat {n}}$

$\mathbf {r} _{i}$ is the shortest vector from the rotation axis to the point mass.

Substituting the formula for $\mathbf {v} _{i}$ into the definition of $\mathbf {L}$ yields

$\mathbf {L} =\sum _{i=1}^{N}m_{i}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {r} _{i})={\boldsymbol {\omega }}\sum _{i=1}^{N}m_{i}r_{i}^{2}=I\omega \mathbf {\hat {n}}$

where we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel): ${\boldsymbol {\omega }}\cdot \mathbf {r} _{i}=0$.

The torque $\mathbf {N}$ is defined as the rate of change of the angular momentum $\mathbf {L}$

$\mathbf {N} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {d\mathbf {L} }{dt}}$

If I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis $\mathbf {\hat {n}}$ so that $\mathrm {I}$ is not changing) then we may write

$\mathbf {N} \ {\stackrel {\mathrm {def} }{=}}\ I{\frac {d\omega }{dt}}\mathbf {\hat {n}} =I\alpha \mathbf {\hat {n}}$

where

$\alpha$ is called the angular acceleration (or rotational acceleration) about the rotation axis $\mathbf {\hat {n}}$.

Notice that if I is not constant in the external reference frame (ie. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torque-free precession.

Conservation of angular momentum allows athletes such as ice skaters, divers, and gymnasts to manipulate their rotation by altering their moment of inertia. For example, consider spinning ice skaters who pull in their arms. Since the ice is nearly frictionless, the angular momentum should stay constant during their spin. When they pull in their arms, the skaters concentrate their mass closer to the rotation axis, decreasing their moment of inertia. To keep the angular momentum constant, the angular velocity $\omega$ increases, resulting in a faster spin.

Moment of inertia tensor

For the same object, different axes of rotation will have different moments of inertia about those axes. For example, the three moments of inertia associated with rotations about the three Cartesian axes (X, Y, and Z),

$I_{xx}=\;$ moment of inertia about the line through the centroid, parallel to the X-axis,

$I_{yy}=\;$ moment of inertia about the line through the centroid, parallel to the Y-axis,

$I_{zz}=\;$ moment of inertia about the line through the centroid, parallel to the Z-axis,

are not guaranteed to be equal unless the object is very symmetric. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity.

Definition

For a rigid object of $N$ point masses $m_{i}$, the moment of inertia tensor is given by

$\mathbf {I} ={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}$.

Its components are defined as

$I_{xx}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=1}^{N}m_{i}(y_{i}^{2}+z_{i}^{2})\,\!$,

$I_{yy}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=1}^{N}m_{i}(x_{i}^{2}+z_{i}^{2})\,\!$,

$I_{zz}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=1}^{N}m_{i}(x_{i}^{2}+y_{i}^{2})\,\!$,

$I_{xy}=I_{yx}\ {\stackrel {\mathrm {def} }{=}}\ -\sum _{i=1}^{N}m_{i}x_{i}y_{i}\,\!$,

$I_{xz}=I_{zx}\ {\stackrel {\mathrm {def} }{=}}\ -\sum _{i=1}^{N}m_{i}x_{i}z_{i}\,\!$ and

$I_{yz}=I_{zy}\ {\stackrel {\mathrm {def} }{=}}\ -\sum _{i=1}^{N}m_{i}y_{i}z_{i}\,\!$,

for Cartesian coordinates $(x_{i},y_{i},z_{i})$, where the origin is the center of mass. Here $I_{xx}$ denotes the moment of inertia around the $x$-axis when the objects are rotated around the x-axis, $I_{xy}$ denotes the moment of inertia around the $y$-axis when the objects are rotated around the $x$-axis, and so on.

These quantities can be generalized to an object with continuous density in a similar fashion to the scalar moment of inertia. We then have

$\mathbf {I} =\int _{V}\left[\left(\mathbf {r} \cdot \mathbf {r} \right)\mathbf {\delta } -\mathbf {r} \otimes \mathbf {r} \right]\ dm$

where $\left(\mathbf {r} \otimes \mathbf {r} \right)_{i,j}=\mathbf {r} _{i}\mathbf {r} _{j}$ and $\mathbf {\delta } \,$ is the 3 x 3 unit matrix.

Reduction to scalar

The scalar form $I$ for any axis $\mathbf {\hat {n}}$ can be calculated from the tensor form $\mathbf {I}$ using the double dot product

$I=\mathbf {\hat {n}} \cdot \mathbf {I} \cdot \mathbf {\hat {n}} =\sum _{j=1}^{3}\sum _{k=1}^{3}n_{j}I_{jk}n_{k}$

where the range of both summations correspond to the three Cartesian coordinates.

Principal moments of inertia

Since this tensor is a symmetric, real matrix, it is possible to find a Cartesian coordinate system in which it is diagonal, i.e., has the form

$\mathbf {I} ={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}$

where the coordinate axes are called the principal axes and the constants $I_{1}$, $I_{2}$ and $I_{3}$ are called the principal moments of inertia and are usually arranged in increasing order

$I_{1}\leq I_{2}\leq I_{3}$

The unit vectors along the principal axes are usually denoted as $(\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})$.

If two principal moments are the same, the rigid body is called a symmetrical top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it need not be spherical) and any axis can be considered a principal axis (since all have equivalent moments of inertia).

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order $m$, i.e., is symmetrical under rotations of 360°/m about a given axis, the symmetry axis is a principal axis. If $m>2$, the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, e.g., a cube or any other Platonic solid.

Parallel axes theorem

If the moment of inertia tensor has been calculated for rotations about the centroid of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the centroid. If the axis of rotation is displaced by a vector $\mathbf {R}$ from the centroid, the new moment of inertia tensor equals

$\mathbf {I} _{jk}^{\mathrm {displaced} }=\mathbf {I} _{jk}^{\mathrm {centroid} }+M\left[\delta _{jk}\,\mathbf {R} \cdot \mathbf {R} -R_{j}R_{k}\right]$

where $M$ is the total mass of the rigid body and $\delta _{jk}$ is the Kronecker delta function.

Other mechanical quantities

Using the tensor $\mathbf {I}$, the kinetic energy can be written as a double dot product

$T={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} \cdot {\boldsymbol {\omega }}={\frac {1}{2}}I_{1}\omega _{1}^{2}+{\frac {1}{2}}I_{2}\omega _{2}^{2}+{\frac {1}{2}}I_{3}\omega _{3}^{2}$

and the angular moment can be written as a single dot product

$\mathbf {L} =\mathbf {I} \cdot {\boldsymbol {\omega }}=\omega _{1}I_{1}\mathbf {e} _{1}+\omega _{2}I_{2}\mathbf {e} _{2}+\omega _{3}I_{3}\mathbf {e} _{3}$

Taken together, we may express the kinetic energy in terms of the angular momentum $(L_{1},L_{2},L_{3})$ in the principal axis frame

$T={\frac {L_{1}^{2}}{2I_{1}}}+{\frac {L_{2}^{2}}{2I_{2}}}+{\frac {L_{3}^{2}}{2I_{3}}}\,\!$

where $L_{k}\ {\stackrel {\mathrm {def} }{=}}\ I_{k}\omega _{k}$ for $k=1,2,3$.

See the article on the rigid rotor for more ways of expressing the kinetic energy of a rigid body.

See also

• List of moments of inertia
• List of moment of inertia tensors
• Torque
• Rotational energy
• Rigid body
• Rigid rotor
• Poinsot's construction

References

• Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).

• Goldstein H. (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9

• Symon KR. (1971) Mechanics, 3rd. ed., Addison-Wesley. ISBN 0-201-07392-7

External links

• Angular momentum and rigid-body rotation in two and three dimensions
• A table of moments of inertia
• Lecture notes on rigid-body rotation and moments of inertia
• The moment of inertia tensor
• An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)