Moment of inertia
Overview
Classical mechanics
In physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI
Si
Si, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units kg·m²) is a measure of an object's resistance to changes to its rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
. It is the inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
and angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
, torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
and angular acceleration
Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha . Mathematical definition :...
, and several other quantities.
Unanswered Questions
Encyclopedia
In classical mechanics
, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI
units kg·m²) is a measure of an object's resistance to changes to its rotation
. It is the inertia
of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum
and angular velocity
, torque
and angular acceleration
, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.
While a simple scalar
treatment of the moment of inertia suffices for many situations, a more advanced tensor
treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.
The concept was introduced by Leonhard Euler
in his book Theoria motus corporum solidorum seu rigidorum in 1765. In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.
(force* distance from axis of rotation) is required to change its rotation rate. For example, consider two hoops, A and B, made of the same material and of equal mass. Hoop A is larger in diameter but thinner than B. It requires more effort to accelerate hoop A (change its angular velocity) because its mass is distributed farther from its axis of rotation: mass that is farther out from that axis must, for a given angular velocity, move more quickly than mass closer in. So in this case, hoop A has a larger moment of inertia than hoop B.
The moment of inertia of an object can change if its shape changes. Figure skaters who begin a spin with arms outstretched provide a striking example. By pulling in their arms, they reduce their moment of inertia, causing them to spin faster (by the conservation of angular momentum
).
The moment of inertia has two forms, a scalar
form, I, (used when the axis of rotation is specified) and a more general tensor
form that does not require the axis of rotation to be specified. The scalar moment of inertia, I, (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of pulleys. For instance, while a block of any shape will slide down a frictionless decline at the same rate, rolling objects may descend at different rates, depending on their moments of inertia. A hoop will descend more slowly than a solid disk of equal mass and radius because more of its mass is located far from the axis of rotation. However, for (more complicated) problems in which the axis of rotation can change, the scalar treatment is inadequate, and the tensor treatment must be used (although shortcuts are possible in special situations). Examples requiring such a treatment include gyroscopes, tops, and even satellites, all objects whose alignment can change.
The moment of inertia is also called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area
, which is sometimes called the area moment of inertia (especially by structural engineers). The easiest way to differentiate these quantities is through their units
(kg·m² as opposed to m^{4}). In addition, moment of inertia should not be confused with polar moment of inertia
(more specifically, polar moment of inertia of area), which is a measure of an object's ability to resist torsion
(twisting) only, although, mathematically, they are similar: if the solid for which the moment of inertia is being calculated has uniform thickness in the direction of the rotating axis, and also has uniform mass density, the difference between the two types of moments of inertia is a factor of mass per unit area.
of the mass is:
Classical mechanics
In physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI
Si
Si, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units kg·m²) is a measure of an object's resistance to changes to its rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
. It is the inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
and angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
, torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
and angular acceleration
Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha . Mathematical definition :...
, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.
While a simple scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or spacetime translations . This is in contrast to a vector...
treatment of the moment of inertia suffices for many situations, a more advanced tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.
The concept was introduced by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in his book Theoria motus corporum solidorum seu rigidorum in 1765. In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.
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. Therefore, it encompasses not just how much mass the object has overall, but how far each bit of mass is from the axis. The farther out the object's mass is, the more rotational inertia the object has, and the more torqueTorque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
(force* distance from axis of rotation) is required to change its rotation rate. For example, consider two hoops, A and B, made of the same material and of equal mass. Hoop A is larger in diameter but thinner than B. It requires more effort to accelerate hoop A (change its angular velocity) because its mass is distributed farther from its axis of rotation: mass that is farther out from that axis must, for a given angular velocity, move more quickly than mass closer in. So in this case, hoop A has a larger moment of inertia than hoop B.
The moment of inertia of an object can change if its shape changes. Figure skaters who begin a spin with arms outstretched provide a striking example. By pulling in their arms, they reduce their moment of inertia, causing them to spin faster (by the conservation of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
).
The moment of inertia has two forms, a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or spacetime translations . This is in contrast to a vector...
form, I, (used when the axis of rotation is specified) and a more general tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
form that does not require the axis of rotation to be specified. The scalar moment of inertia, I, (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of pulleys. For instance, while a block of any shape will slide down a frictionless decline at the same rate, rolling objects may descend at different rates, depending on their moments of inertia. A hoop will descend more slowly than a solid disk of equal mass and radius because more of its mass is located far from the axis of rotation. However, for (more complicated) problems in which the axis of rotation can change, the scalar treatment is inadequate, and the tensor treatment must be used (although shortcuts are possible in special situations). Examples requiring such a treatment include gyroscopes, tops, and even satellites, all objects whose alignment can change.
The moment of inertia is also called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area
Second moment of area
The second moment of area, also known as the area moment of inertia, moment of inertia of plane area, or second moment of inertia is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the crosssectional plane...
, which is sometimes called the area moment of inertia (especially by structural engineers). The easiest way to differentiate these quantities is through their units
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
(kg·m² as opposed to m^{4}). In addition, moment of inertia should not be confused with polar moment of inertia
Polar moment of inertia
Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects with an invariant circular cross section and no significant warping or outofplane deformation. It is used to calculate the angular displacement of an object subjected to a torque...
(more specifically, polar moment of inertia of area), which is a measure of an object's ability to resist torsion
Torsion (mechanics)
In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....
(twisting) only, although, mathematically, they are similar: if the solid for which the moment of inertia is being calculated has uniform thickness in the direction of the rotating axis, and also has uniform mass density, the difference between the two types of moments of inertia is a factor of mass per unit area.
Scalar moment of inertia for single body
Consider a massless rigid rod of length with a point mass at one end and rotating about the other end. Suppose the rod rotates at a constant rate so that the mass moves at speed . Then the kinetic energyKinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
of the mass is:

Using , where ω is the angular velocityAngular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
, one obtains:

which can be rearranged to give:

This equation resembles the original expression for the kinetic energy, but in the place of the linear velocity is the angular velocity , and instead of the mass is . The quantity can therefore be seen as an analogue of mass for rotational motion; in other words, it is a measure of rotational inertia.
Scalar moment of inertia for many bodies
Consider a rigid bodyRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
rotating with angular velocityAngular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
ω around a certain axis. The body consists of N point masses m_{i} whose distances to the axis of rotation are denoted r_{i}. Each point mass will have the speed , so that the total kinetic energyKinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
T of the body can be calculated as

In this expression the quantity in parentheses is called the moment of inertia of the body (with respect to the specified axis of rotation). It is a purely geometric characteristic of the object, as it depends only on its shape and the position of the rotation axis. The moment of inertia is usually denoted with the capital letter I:

It is worth emphasizing that r_{i} here is the distance from a point to the axis of rotation, not to the origin. As such, the moment of inertia will be different when considering rotations about different axes.
Similarly, the moment of inertia of a continuous solid body rotating about a known axis can be calculated by replacing the summation with the integralMultiple integralThe multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...
:

where r is the radius vector of a point within the body, ρ(r) is the mass densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
at point r, and d(r) is the distance from point r to the axis of rotation. The integration is evaluated over the volume V of the body.
Moment of Inertia about a point
In discussions of the virial theorem, a different definition of moment of inertia is introduced, often qualified as moment of inertia about a point (as opposed to an axis). The defining equations look the same as the ones above, but and in these cases are understood to be the distance to the origin.
Moment of Inertia Theorems
Calculations for the Moment of Inertia (MOI) of a body are not easy in general. The process can be simplified in the following ways:
 Choosing axes to take advantage of geometric symmetries
 Physical homeogeneity (i.e. uniform mass distribution) making the density function ρ(r) become constant (elementary calculations, or generally an approximation)
 Use of the following theorems , see below for details.
Theorem Nomenclature Equation Superposition Principle for
MOI about any chosen AxisI_{net} = Resultant MOI (about any one axis) Parallel axis theorem Parallel axis theoremIn physics, the parallel axis theorem or HuygensSteiner 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...M = Total mass of body
d = Perpendicular distance from an axis
through the COM to another parallel axis
I_{COM} = MOI about the axis through
the COM
I = MOI about the parallel axisPerpendicular axis theorem Perpendicular axis theoremIn physics, the perpendicular axis theorem can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane...i, j, k refer to MOI about any three mutually
perpendicular axes.
The sum of MOI about any two is greater than
or equal to the third.
Properties
Superposition
The moment of inertia of the body is additive. That is, if a body can be decomposed (either physically or conceptually) into several constituent parts, then the moment of inertia of the whole body about a given axis is equal to the sum of moments of inertia of each part around the same axis.
Basing just on the dimensional analysisDimensional analysisIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
, the moment of inertia must take the form , where M is the mass, L is the “size” of the body in the direction perpendicular to the axis of rotation, and c is a dimensionless inertial constant. Additionally, the length $\backslash scriptstyle\; R\backslash ,=\backslash ,\backslash sqrt\{I/M\}$ is called the radius of gyrationRadius of gyrationRadius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....
of the body.
Perpendicular Axes
If I_{x}, I_{y}, I_{z} are moments of inertia around three perpendicular axes passing through the body’s center of mass, then each of them cannot be greater than the sum of two others: for example . Here the equality holds only if the body is flat and located in the Oxy coordinate plane.
Parallel Axes
If the object’s moment of inertia I_{COM} around a certain axis passing through the center of mass is known, then the parallel axis theoremParallel axis theoremIn physics, the parallel axis theorem or HuygensSteiner 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...
or Huygens–Steiner theorem provides a convenient formula to compute the moment of inertia I_{d} of the same body around a different axis, which is parallel to the original and located at a distance d from it. The formula is only suitable when the initial and final axes are parallel. In order to compute the moment of inertia about an arbitrary axis, one has to use the object’s moment of inertia tensor.
Energy, Angular Momentum, Torque
The rotational kinetic energy of a rigid bodyRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
with angular velocity ω (in radianRadianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s per second) is expressed in terms of the object’s moment of inertia:

This formula is similar to the translational kinetic energy . Thus, the moment of inertia I plays the role of mass in rotational dynamics. One key difference between the mass and the (scalar) moment of inertia is that the latter depends on the axis of rotation and so is not truly invariant. The invariant characteristic of the body in rotational motion is the tensor of moment of inertia I, defined later.
The angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the body rotating around one of its principal axes is also proportional to the moment of inertia:

This expression is parallel to the formula for translational momentumMomentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, where the moment of inertia I plays the role of mass m, and the angular velocity ω stands for the velocity v. The scalar formula is valid for rotations around one of the principal axes of the body or for rotation about any fixed axis. The equivalent formula involving the tensor moment of inertia is always correct and must be used in cases of free rotation about a nonprincipal axis.
Also when the body rotates around one of its principal axes, and the direction of the axis of rotation remains constant, one can relate the torqueTorqueTorque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
on an object and its angular accelerationAngular accelerationAngular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha . Mathematical definition :...
in a similar equation:

where τ is the torque and α is the angular accelerationAngular accelerationAngular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha . Mathematical definition :...
.
Examples
Diatomic molecule, with atoms m_{1} and m_{2} at a distance d from each other, rotating around the axis which passes through the molecule’s center of massCenter of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
and is perpendicular to the direction of the molecule.
The easiest way to calculate this molecule’s moment of inertia is to use the parallel axis theoremParallel axis theoremIn physics, the parallel axis theorem or HuygensSteiner 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...
. If we consider rotation around the axis passing through the atom m_{1}, then the moment of inertia will be . On the other hand, by the parallel axis theorem this moment is equal to , where I is the moment of inertia around the axis passing through the center of mass, and a is the distance between the center of mass and the first atom. By the center of massCenter of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
formula, this distance is equal to . Thus,

Thin rod of mass m and length ℓ, rotating around the axis which passes through its center and is perpendicular to the rod.
Let Oz be the axis of rotation, and Ox the axis along the rod. If ρ is the density, and s the crosssection of the rod (so that ), then the volume element for the integral formula will be equal to , where x changes from −½ℓ to ½ℓ. The moment of inertia can be found by computing the integral:

Solid ballBall (mathematics)In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
of mass m and radius R, rotating around an axis which passes through the center.
Suppose Oz is the axis of rotation. The distance from point towards the axis Oz is equal to . Thus, in order to compute the moment of inertia I_{z}, we need to evaluate the integral . The calculation considerably simplifies if we notice that by symmetry of the problem, the moments of inertia around all axes are equal: . Then

where is the distance from point r to the origin. This integral is easy to evaluate in the spherical coordinates, the volume element will be equal to , where r goes from 0 to R. Thus,

In mechanicsMechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
of machineMachineA machine manages power to accomplish a task, examples include, a mechanical system, a computing system, an electronic system, and a molecular machine. In common usage, the meaning is that of a device having parts that perform or assist in performing any type of work...
s, when designing rotary parts like gears, pulleys, shafts, couplings etc., which are used to transmit torques, the moment of inertia has to be considered. The moment of inertia is given about an axis and it depends on the shape, densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of a rotating element.
When considering mechanisms like gear trainGear trainA gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next.The transmission of...
s, worm and wheel, where there are more than one rotating element, more than one axis of rotation, an equivalent moment of inertia for the system should be found. Practically when a geared system is enclosed, equivalent moment of inertia can be measured by measuring the angular acceleration for a known torque or theoretically it can be estimated when the masses and dimensions of the rotating elements and shafts are known. In this practical the equivalent moment of inertia of a worm and wheel system is measured using above mention methods.
Moment of inertia tensor
In three dimensions, if the axis of rotation is not given, we need to be able to generalize the scalar moment of inertia to a quantity that allows us to compute a moment of inertia about arbitrary axes. This quantity is known as the moment of inertia tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
and can be represented as a symmetric positive semidefinite matrix, I. This representation elegantly generalizes the scalar case: The angular momentum vector is related to the rotation velocity vector ω by
and the kinetic energy is given by
as compared with
in the scalar case.
Like the scalar moment of inertia, the moment of inertia tensor may be calculated with respect to any point in space, but for practical purposes, the center of mass is almost always used. In general, its components are time dependent.
Definition
For a rigid object of point masses , the moment of inertia tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
(with respect to the origin) has components given by
 ,
where
and , , and .
(Thus is a symmetric tensor.) Note that the scalars with are called the products of inertia.
Here denotes the moment of inertia around the axis when the objects are rotated around the xaxis, denotes the moment of inertia around the axis when the objects are rotated around the 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
In moreconcise notation:
where is the vector from the center of mass to a point in the volume, V and is their outer productOuter productIn linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
, E_{3} is the identity matrixIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, and V is a region of space completely containing the object. Alternatively, the above can be described in Einstein notationEinstein notationIn mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
as:
where is the Kronecker delta.
The diagonal elements of are called the principal moments of inertia.
As is clear from the definition, and its components are in general timedependent. They are constant only if calculated in a reference frame moving rigidly with the rigid body. On the other hand, the eigenvalues are invariant, that is independent of the reference frame used to calculate .
Derivation of the tensor components
The distance of a particle at from the axis of rotation passing through the origin in the direction is
. By using the formula (and some simple vector algebra) it can be seen that the moment of inertia of this particle (about the axis of rotation passing through the origin in the direction) is
This is a quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy  3y^2\,\!is a quadratic form in the variables x and y....
in and, after a bit more algebra, this leads to a tensor formula for the moment of inertia
 .
This is exactly the formula given below for the moment of inertia in the case of a single particle. For multiple particles we need only recall that the moment of inertia is additive in order to see that this formula is correct.
Reduction to scalar
For any axis , represented as a column vector with elements n_{i}, the scalar form I can be calculated from the tensor form I as

The range of both summations correspond to the three Cartesian coordinates.
The following equivalent expression avoids the use of transposed vectors which are not supported in maths libraries because internally vectors and their transpose are stored as the same linear array,

However it should be noted that although this equation is mathematically equivalent to the equation above for any matrix, inertia tensors are symmetrical. This means that it can be further simplified to:

Principal axes of inertia
By the spectral theoremSpectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
, since the moment of inertia tensor is real and symmetric, there exists a Cartesian coordinate system in which it is diagonalDiagonalizable matrixIn linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
, having the form

where the coordinate axes are called the principal axes and the
constants , and are called the principal moments of inertia.
This result was first shown by , and is a form of Sylvester's law of inertiaSylvester's law of inertiaSylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...
. 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 massCenter of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
are uniquely specified. 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, 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 , i.e., is symmetrical under rotations of about a given axis, the symmetry axis is a principal axis. When , 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 solidPlatonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
.
The motionMotion (physics)In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...
of vehicleVehicleA vehicle is a device that is designed or used to transport people or cargo. Most often vehicles are manufactured, such as bicycles, cars, motorcycles, trains, ships, boats, and aircraft....
s is often described about these axes with the rotationRotationA rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s called yaw, pitch, and roll.
A practical example of this mathematical phenomenon is the routine automotive task of balancing a tireTire balanceTire balance, also referred to as tire unbalance or imbalance, describes the distribution of mass within an automobile tire or the wheel to which it is attached. When the tire rotates, asymmetries of mass cause the wheel to wobble, which can cause ride disturbances, usually vertical and lateral...
, 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.
Parallel axis theorem
Once the moment of inertia tensor has been calculated for rotations about the center of massCenter of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
of the rigid body, there is a useful laborsaving method to compute the tensor for rotations offset from the center of mass.
If the axis of rotation is displaced by a vector R from the center of mass, the new moment of inertia tensor equals

where m is the total mass of the rigid body, E_{3} is the identity matrixIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, and is the outer productOuter productIn linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
.
Rotational symmetry
Using the above equation to express all moments of inertia in terms of integrals of variables either along or perpendicular to the axis of symmetry
usually simplifies the calculation of these moments considerably.
Comparison with covariance matrix
The moment of inertia tensor about the center of mass of a 3 dimensional rigid body is related to the covariance matrixCovariance matrixIn probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
of a trivariate random vector whose probability density functionProbability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
is proportional to the pointwise density of the rigid body by:
where n is the number of points.
The structure of the momentofinertia tensor comes from the fact that it is to be used as a bilinear form on rotation vectors in the form
Each element of mass has a kinetic energy of
The velocity of each element of mass is where r is a vector from the center of rotation to that element of mass. The cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in threedimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
can be converted to matrix multiplication so that
and similarly
Thus,
plugging in the definition of the term leads directly to the structure of the moment tensor.
See also
 List of moments of inertia
 List of moment of inertia tensors
 Rotational energyRotational energyThe rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy...
 Parallel axis theoremParallel axis theoremIn physics, the parallel axis theorem or HuygensSteiner 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...
 Perpendicular axis theoremPerpendicular axis theoremIn physics, the perpendicular axis theorem can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane...
 Stretch ruleStretch ruleIn physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation,...
 Tire balanceTire balanceTire balance, also referred to as tire unbalance or imbalance, describes the distribution of mass within an automobile tire or the wheel to which it is attached. When the tire rotates, asymmetries of mass cause the wheel to wobble, which can cause ride disturbances, usually vertical and lateral...
 Poinsot's ellipsoid
 Instant centre of rotationInstant centre of rotationThe instant centre of rotation, also called instantaneous centre and instant centre, is the point in a body undergoing planar movement that has zero velocity at a particular instant of time...
External links
 Angular momentum and rigidbody rotation in two and three dimensions
 Lecture notes on rigidbody 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)
 Tutorial on finding moments of inertia, with problems and solutions on various basic shapes
 Testing Facilities for the Moment of Inertia Rig at Cranfield Impact Centre















