Moment of inertia

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
Introduction.
When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared (kg·m2) in SI units and pound-foot-second squared (lbf·ft·s2) in imperial or US units.
Moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by formula_1, where formula_2 is the distance of the point from the axis, and formula_3 is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. The term "moment of inertia" was introduced by Leonhard Euler in his book "Theoria motus corporum solidorum seu rigidorum" in 1765, and it is incorporated into Euler's second law.
The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.
Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.
The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.
Definition.
Moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section . 
Moment of inertia is defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is
If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster.
If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is
For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as,
Thus, moment of inertia of the pendulum depends on both the mass of a body and its geometry, or shape, as defined by the distance to the axis of rotation.
This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.
In general, given an object of mass , an effective radius can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is
where is known as the radius of gyration around the axis.
Examples.
Simple pendulum.
Moment of inertia can be measured using a simple pendulum, because it is the resistance to the rotation caused by gravity. Mathematically, the moment of inertia of the pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle formula_3 with the square of its distance formula_2 to the pivot, that is
This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torque formula_11 around the axis perpendicular to the plane of the pendulum movement. Here formula_12 is the distance vector perpendicular to and from the force to the torque axis, and formula_13 is the net force on the mass. Associated with this torque is an angular acceleration, formula_14, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is formula_15. Since formula_16 the torque equation becomes:
where formula_18 is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of formula_14 and formula_12.) The quantity formula_21 is the "moment of inertia" of this single mass around the pivot point.
The quantity formula_21 also appears in the angular momentum of a simple pendulum, which is calculated from the velocity formula_23 of the pendulum mass around the pivot, where formula_24 is the angular velocity of the mass about the pivot point. This angular momentum is given by
using a similar derivation to the previous equation.
Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
This shows that the quantity formula_21 is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values formula_1 for all of the elements of mass in the body.
Compound pendulum.
A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency (formula_29) of a compound pendulum depends on its moment of inertia, formula_30,
where formula_3 is the mass of the object, formula_33 is local acceleration of gravity, and formula_2 is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point formula_35 so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (formula_36), to obtain
where formula_36 is the period (duration) of oscillation (usually averaged over multiple periods).
Center of oscillation.
A simple pendulum that has the same natural frequency as a compound pendulum defines the length formula_39 from the pivot to a point called the center of oscillation of the compound pendulum. This point also corresponds to the center of percussion. The length formula_39 is determined from the formula,
or
The seconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of formula_43 for the pendulum. In this case, the distance to the center of oscillation, formula_39, can be computed to be
Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.
Measuring moment of inertia.
The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
Motion in a fixed plane.
Point mass.
The moment of inertia about an axis of a body is calculated by summing formula_1 for every particle in the body, where formula_2 is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.)
Consider the kinetic energy of an assembly of formula_48 masses formula_49 that lie at the distances formula_50 from the pivot point formula_35, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,
This shows that the moment of inertia of the body is the sum of each of the formula_1 terms, that is
Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:
Another expression replaces the summation with an integral,
Here, the function formula_57 gives the mass density at each point formula_58, formula_12 is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point formula_58 in the solid, and the integration is evaluated over the volume formula_61 of the body formula_62. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the formula_63-axis perpendicular to the cross-section, weighted by its density. This is also called the "polar moment of the area", and is the sum of the second moments about the formula_64- and formula_65-axes. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the formula_64-axis or formula_65-axis depending on the load.
Examples.
The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.
 = \iiint\limits_Q \rho\,x^2\,\mathrm{d}V =
 \int_{-\frac{\ell}{2}}^\frac{\ell}{2} \rho\,x^2 s\,\mathrm{d}x =
 \left. \rho s\frac{x^3}{3}\right|_{-\frac{\ell}{2}}^\frac{\ell}{2} =
 \frac{\rho s}{3} \left(\frac{\ell^3}{8} + \frac{\ell^3}{8}\right) =
 \frac{m\ell^2}{12},
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where formula_68 is the mass of the rod.
where formula_75 is its mass.
where formula_39 is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the center of mass to the pivot point of the pendulum.
A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly.
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that form the sphere. If the surface of the ball is defined by the equation
then the square of the radius formula_2 of the disc at the cross-section formula_63 along the formula_63-axis is
Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the formula_63-axis,
where formula_86 is the mass of the sphere.
Rigid body.
If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis formula_18 perpendicular to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the "polar moment of inertia". The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.
If a system of formula_88 particles, formula_89, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point formula_90, and absolute velocities formula_91:
where formula_24 is the angular velocity of the system and formula_94 is the velocity of formula_90.
For planar movement the angular velocity vector is directed along the unit vector formula_96 which is perpendicular to the plane of movement. Introduce the unit vectors formula_97 from the reference point formula_90 to a point formula_99, and the unit vector formula_100, so
This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.
Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.
Angular momentum.
The angular momentum vector for the planar movement of a rigid system of particles is given by
Use the center of mass formula_103 as the reference point so
and define the moment of inertia relative to the center of mass formula_105 as
then the equation for angular momentum simplifies to
The moment of inertia formula_105 about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the "polar moment of inertia". Specifically, it is the second moment of mass with respect to the orthogonal distance from an axis (or pole).
For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.
Kinetic energy.
The kinetic energy of a rigid system of particles moving in the plane is given by
Let the reference point be the center of mass formula_103 of the system so the second term becomes zero, and introduce the moment of inertia formula_105 so the kinetic energy is given by
The moment of inertia formula_105 is the "polar moment of inertia" of the body.
Newton's laws.
Newton's laws for a rigid system of formula_88 particles, formula_89, can be written in terms of a resultant force and torque at a reference point formula_90, to yield
where formula_99 denotes the trajectory of each particle.
The kinematics of a rigid body yields the formula for the acceleration of the particle formula_119 in terms of the position formula_90 and acceleration formula_121 of the reference particle as well as the angular velocity vector formula_24 and angular acceleration vector formula_14 of the rigid system of particles as,
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along formula_18 perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors formula_126 from the reference point formula_90 to a point formula_99 and the unit vectors formula_100, so
This yields the resultant torque on the system as
where formula_132, and formula_133 is the unit vector perpendicular to the plane for all of the particles formula_119.
Use the center of mass formula_103 as the reference point and define the moment of inertia relative to the center of mass formula_105, then the equation for the resultant torque simplifies to
Motion in space of a rigid body, and the inertia matrix.
The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.
Let the system of formula_88 particles, formula_89 be located at the coordinates formula_99 with velocities formula_91 relative to a fixed reference frame. For a (possibly moving) reference point formula_90, the relative positions are
and the (absolute) velocities are
where formula_24 is the angular velocity of the system, and formula_146 is the velocity of formula_90.
Angular momentum.
Note that the cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, formula_148, constructed from the components of formula_149:
The inertia matrix is constructed by considering the angular momentum, with the reference point formula_90 of the body chosen to be the center of mass formula_103:
where the terms containing formula_146 (formula_155) sum to zero by the definition of center of mass.
Then, the skew-symmetric matrix formula_156 obtained from the relative position vector formula_157, can be used to define,
where formula_159 defined by
is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass formula_103.
Kinetic energy.
The kinetic energy of a rigid system of particles can be formulated in terms of the center of mass and a matrix of mass moments of inertia of the system. Let the system of formula_88 particles formula_89 be located at the coordinates formula_99 with velocities formula_91, then the kinetic energy is
where formula_157 is the position vector of a particle relative to the center of mass.
This equation expands to yield three terms
The second term in this equation is zero because formula_103 is the center of mass. Introduce the skew-symmetric matrix formula_156 so the kinetic energy becomes
Thus, the kinetic energy of the rigid system of particles is given by
where formula_159 is the inertia matrix relative to the center of mass and formula_174 is the total mass.
Resultant torque.
The inertia matrix appears in the application of Newton's second law to a rigid assembly of particles. The resultant torque on this system is,
where formula_176 is the acceleration of the particle formula_119. The kinematics of a rigid body yields the formula for the acceleration of the particle formula_119 in terms of the position formula_90 and acceleration formula_180 of the reference point, as well as the angular velocity vector formula_24 and angular acceleration vector formula_14 of the rigid system as,
Use the center of mass formula_103 as the reference point, and introduce the skew-symmetric matrix formula_185 to represent the cross product formula_186, to obtain
The calculation uses the identity
obtained from the Jacobi identity for the triple cross product as shown in the proof below:
Thus, the resultant torque on the rigid system of particles is given by
where formula_159 is the inertia matrix relative to the center of mass.
Parallel axis theorem.
The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass formula_103 and the inertia matrix relative to another point formula_90. This relationship is called the parallel axis theorem.
Consider the inertia matrix formula_193 obtained for a rigid system of particles measured relative to a reference point formula_90, given by
Let formula_103 be the center of mass of the rigid system, then
where formula_198 is the vector from the center of mass formula_103 to the reference point formula_90. Use this equation to compute the inertia matrix,
Distribute over the cross product to obtain
The first term is the inertia matrix formula_159 relative to the center of mass. The second and third terms are zero by definition of the center of mass formula_103. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix formula_205 constructed from formula_198.
The result is the parallel axis theorem,
where formula_198 is the vector from the center of mass formula_103 to the reference point formula_90.
Note on the minus sign: By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form formula_211, which is similar to the formula_1 that appears in planar movement. However, to make this to work out correctly a minus sign is needed. This minus sign can be absorbed into the term formula_213, if desired, by using the skew-symmetry property of formula_214.
Scalar moment of inertia in a plane.
The scalar moment of inertia, formula_215, of a body about a specified axis whose direction is specified by the unit vector formula_18 and passes through the body at a point formula_90 is as follows:
where formula_193 is the moment of inertia matrix of the system relative to the reference point formula_90, and formula_156 is the skew symmetric matrix obtained from the vector formula_143.
This is derived as follows. Let a rigid assembly of formula_88 particles, formula_89, have coordinates formula_99. Choose formula_90 as a reference point and compute the moment of inertia around a line L defined by the unit vector formula_18 through the reference point formula_90, formula_229. The perpendicular vector from this line to the particle formula_119 is obtained from formula_231 by removing the component that projects onto formula_18.
where formula_234 is the identity matrix, so as to avoid confusion with the inertia matrix, and formula_235 is the outer product matrix formed from the unit vector formula_18 along the line formula_39.
To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix formula_238 such that formula_239, then we have the identity
noting that formula_18 is a unit vector.
The magnitude squared of the perpendicular vector is
The simplification of this equation uses the triple scalar product identity
where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting that formula_231 and formula_18 are orthogonal:
Thus, the moment of inertia around the line formula_39 through formula_90 in the direction formula_18 is obtained from the calculation
where formula_193 is the moment of inertia matrix of the system relative to the reference point formula_90.
This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.
Inertia tensor.
For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.
Definition.
For a rigid object of formula_48 point masses formula_254, the moment of inertia tensor is given by
Its components are defined as
where
Note that, by the definition, formula_265 is a symmetric tensor.
The diagonal elements are more succinctly written as
while the off-diagonal elements, also called the products of inertia, are
Here formula_268 denotes the moment of inertia around the formula_64-axis when the objects are rotated around the x-axis, formula_270 denotes the moment of inertia around the formula_65-axis when the objects are rotated around the formula_64-axis, and so on.
These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has
where formula_274 is their outer product, E3 is the 3 × 3 identity matrix, and "V" is a region of space completely containing the object.
Alternatively it can also be written in terms of the angular momentum operator formula_275:
The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction formula_277,
where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as formula_279 is obtained by the computation
and can be interpreted as the moment of inertia around the formula_64-axis when the object rotates around the formula_65-axis.
The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
It is common in rigid body mechanics to use notation that explicitly identifies the formula_64, formula_65, and formula_63-axes, such as formula_268 and formula_270, for the components of the inertia tensor.
Alternate inertia convention.
There are some CAD and CAE applications such as SolidWorks, Unigraphics NX/Siemens NX and MSC Adams that use an alternate convention for the products of inertia. According to this convention, the minus sign is removed from the product of inertia formulas and instead inserted in the inertia matrix:
Derivation of the tensor components.
The distance formula_2 of a particle at formula_291 from the axis of rotation passing through the origin in the formula_292 direction is formula_293, where formula_292 is unit vector. The moment of inertia on the axis is
Rewrite the equation using matrix transpose:
where E3 is the 3 × 3 identity matrix.
This leads to a tensor formula for the moment of inertia
For multiple particles, we need only recall that the moment of inertia is additive in order to see that this formula is correct.
Inertia tensor of translation.
Let formula_298 be the inertia tensor of a body calculated at its centre of mass, and formula_90 be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by:
where formula_3 is the body's mass, E3 is the 3 × 3 identity matrix, and formula_302 is the outer product.
Inertia tensor of rotation.
Let formula_90 be the matrix that represents a body's rotation. The inertial tensor of the rotated body is given by:
Inertia matrix in different reference frames.
The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame. This means that as the body moves the components of the inertia matrix change with time. In contrast, the components of the inertia matrix measured in a body-fixed frame are constant.
Body frame.
Let the body frame inertia matrix relative to the center of mass be denoted formula_305, and define the orientation of the body frame relative to the inertial frame by the rotation matrix formula_121, such that,
where vectors formula_308 in the body fixed coordinate frame have coordinates formula_291 in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by
Notice that formula_121 changes as the body moves, while formula_305 remains constant.
Principal axes.
Measured in the body frame the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix formula_313 and a diagonal matrix formula_314, given by
where
The columns of the rotation matrix formula_313 define the directions of the principal axes of the body, and the constants formula_318, formula_319, and formula_320 are called the principal moments of inertia. This result was first shown by J. J. Sylvester (1852), and is a form of Sylvester's law of inertia. The principal axis with the highest moment of inertia is sometimes called the figure axis or axis of figure.
When all principal moments of inertia are distinct, the principal axes through center of mass are uniquely specified and the rigid body is called an asymmetric top. If two principal moments are the same, the rigid body is called a symmetric 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, meaning that the moment of inertia is the same about any axis.
The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order formula_3, meaning it is symmetrical under rotations of about the given axis, that axis is a principal axis. When formula_322, the rigid body is a symmetric 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, for example, a cube or any other Platonic solid.
The motion of vehicles is often described in terms of yaw, pitch, and roll which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.
A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.
Rotating molecules are also classified as asymmetric, symmetric or spherical tops, and the structure of their rotational spectra is different for each type.
Ellipsoid.
The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called Poinsot's ellipsoid. Let formula_314 be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface
or
defines an ellipsoid in the body frame. Write this equation in the form,
to see that the semi-principal diameters of this ellipsoid are given by
Let a point formula_291 on this ellipsoid be defined in terms of its magnitude and direction, formula_329, where formula_277 is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia formula_331 around an axis in the direction formula_277, yields
Thus, the magnitude of a point formula_291 in the direction formula_277 on the inertia ellipsoid is