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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 In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
is a quantity that determines the
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
needed for a desired angular acceleration about a rotational axis, akin to how
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
determines the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
needed for a desired
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
. 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 A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
the moment of inertia is simply the mass times the square of the
perpendicular distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. Th ...
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 An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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 In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
and torques around the axes act independently of each other.


Introduction

When a body is free to rotate around an axis,
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
must be applied to change its
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. The amount of torque needed to cause any given angular acceleration (the rate of change in
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
) is proportional to the moment of inertia of the body. Moments 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 Imperial is that which relates to an empire, emperor, or imperialism. Imperial or The Imperial may also refer to: Places United States * Imperial, California * Imperial, Missouri * Imperial, Nebraska * Imperial, Pennsylvania * Imperial, Texa ...
or US units. The moment of inertia plays the role in rotational kinetics that
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
(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 mr^2, where r is the distance of the point from the axis, and m 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 Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a
compound pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
. The term ''moment of inertia'' was introduced by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in his book ''Theoria motus corporum solidorum seu rigidorum'' in 1765, From page 166: ''"Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur."'' (Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.) 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. The moment of inertia also appears in
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
,
kinetic energy In physics, the kinetic energy of an object is the energy that 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 accele ...
, and in
Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
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 Ã— 3 matrix of moments of inertia, called the inertia matrix or inertia tensor. The moment of inertia of a rotating
flywheel A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, ass ...
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

The 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
of the section. The moment of inertia is also defined as the ratio of the net
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of a system to its
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
around a principal axis, that is I = \frac. 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 skater Figure skating is a sport in which individuals, pairs, or groups perform on figure skates on ice. It was the first winter sport to be included in the Olympic Games, when contested at the 1908 Olympics in London. The Olympic disciplines are me ...
s pull in their outstretched arms or divers curl their bodies into a tuck position during a dive. If the shape of the body does not change, then its moment of inertia appears in
Newton's law of motion Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is \tau = I \alpha. For a
simple pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, 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, I = mr^2. Thus, the 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 I = m k^2, where is known as the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
around the axis.


Examples


Simple pendulum

Mathematically, the moment of inertia of a simple 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 m with the square of its distance r to the pivot, that is I = mr^2. This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torque \boldsymbol = \mathbf \times \mathbf around the axis perpendicular to the plane of the pendulum movement. Here \mathbf is the distance vector from the torque axis to the pendulum center of mass, and \mathbf is the net force on the mass. Associated with this torque is an angular acceleration, \boldsymbol, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is \mathbf = \boldsymbol \times \mathbf. Since \mathbf F = m \mathbf a the torque equation becomes: \begin \boldsymbol &= \mathbf \times \mathbf = \mathbf \times (m \boldsymbol \times \mathbf) \\ &= m \left(\left(\mathbf \cdot \mathbf\right) \boldsymbol - \left(\mathbf \cdot \boldsymbol\right) \mathbf\right) \\ &= mr^2 \boldsymbol = I\alpha \mathbf, \end where \mathbf 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 \boldsymbol and \mathbf.) The quantity I = mr^2 is the ''moment of inertia'' of this single mass around the pivot point. The quantity I = mr^2 also appears in the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of a simple pendulum, which is calculated from the velocity \mathbf = \boldsymbol \times \mathbf of the pendulum mass around the pivot, where \boldsymbol is the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of the mass about the pivot point. This angular momentum is given by \begin \mathbf &= \mathbf \times \mathbf = \mathbf \times \left(m\boldsymbol \times \mathbf\right) \\ & = m\left(\left(\mathbf \cdot \mathbf\right)\boldsymbol - \left(\mathbf \cdot \boldsymbol\right)\mathbf\right) \\ &= mr^2 \boldsymbol = I\omega\mathbf, \end 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 E_\text = \frac m \mathbf \cdot \mathbf = \frac \left(mr^2\right)\omega^2 = \fracI\omega^2. This shows that the quantity I = mr^2 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 mr^2 for all of the elements of mass in the body.


Compound pendulums

A
compound pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
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 Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(\omega_\text) of a compound pendulum depends on its moment of inertia, I_P, \omega_\text = \sqrt, where m is the mass of the object, g is local acceleration of gravity, and r 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 P 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 (t), to obtain I_P = \frac = \frac, where t 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 L from the pivot to a point called the
center of oscillation The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
of the compound pendulum. This point also corresponds to the center of percussion. The length L is determined from the formula, \omega_\text = \sqrt = \sqrt, or L = \frac = \frac. The
seconds pendulum A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz. Pendulum A pendulum is a weight suspended from a pivot so that ...
, 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 \pi \ \mathrm for the pendulum. In this case, the distance to the center of oscillation, L, can be computed to be L = \frac \approx \frac \approx 0.99 \ \mathrm. 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 A Kater's pendulum is a reversible free swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity. Its advantage is that, unlike previous ...
is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a
gravimeter Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest. Units of measurement Gr ...
.


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 pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
. 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.


Moment of inertia of area

Moment of inertia of area is also known as the
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
. These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis I_ and horizontal moment of the y-axis I_.
Height (''h'') and breadth (''b'') are the linear measures, except for circles, which are effectively half-breadth derived, r


Sectional areas moment calculated thus

# Square: I_=I_=\frac
# Rectangular: I_=\frac and; I_=\frac
# Triangular: I_=\frac
# Circular: I_=I_=\frac r^4


Motion in a fixed plane


Point mass

The moment of inertia about an axis of a body is calculated by summing mr^2 for every particle in the body, where r 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 N masses m_i that lie at the distances r_i from the pivot point P, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses, E_\text = \sum_^N \frac\,m_i \mathbf_i \cdot \mathbf_i = \sum_^N \frac\,m_i \left(\omega r_i\right)^2 = \frac12\, \omega^2 \sum_^N m_i r_i^2. This shows that the moment of inertia of the body is the sum of each of the mr^2 terms, that is I_P = \sum_^N m_i r_i^2. 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: I_P = \sum_i m_i r_i^2 Another expression replaces the summation with an
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, I_P = \iiint_ \rho(x, y, z) \left\, \mathbf\right\, ^2 dV Here, the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
\rho gives the mass density at each point (x, y, z), \mathbf is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point (x, y, z) in the solid, and the integration is evaluated over the volume V of the body Q. 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 The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
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 z-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 x- and y-axes. The stresses in a
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized grou ...
are calculated using the second moment of the cross-sectional area around either the x-axis or y-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. * The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. Align the x-axis with the rod and locate the origin its center of mass at the center of the rod, then I_ = \iiint_Q \rho\,x^2 \, dV = \int_^\frac \rho\,x^2 s\, dx = \left. \rho s\frac\_^\frac = \frac \left(\frac + \frac\right) = \frac, where m = \rho s \ell is the mass of the rod. * The moment of inertia of a thin disc of constant thickness s, radius R, and density \rho about an axis through its center and perpendicular to its face (parallel to its axis of
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
) is determined by integration. Align the z-axis with the axis of the disc and define a volume element as dV = sr \, dr\, d\theta, then I_ = \iiint_Q \rho \, r^2\, dV = \int_0^ \int_0^R \rho r^2 s r\, dr\, d\theta = 2\pi \rho s \frac = \fracmR^2, where m = \pi R^2 \rho s is its mass. * The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point P as, I_P = I_ + M_\text\left(\frac\right)^2 + I_ + M_\text(L + R)^2, where L 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 Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
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 The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body abo ...
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 can form the sphere whose centers are along the axis chosen for consideration. If the surface of the ball is defined by the equation x^2 + y^2 + z^2 = R^2, then the square of the radius r of the disc at the cross-section z along the z-axis is r(z)^2 = x^2 + y^2 = R^2 - z^2. Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the z-axis, \begin I_ &= \int_^R \frac r(z)^4\, dz = \int_^R \frac \left(R^2 - z^2\right)^2\,dz \\ &= \frac \left ^4z - \frac R^2 z^3 + \frac z^5\right^R \\ &= \pi \rho\left(1 - \frac + \frac\right)R^5 \\ &= \fracmR^2, \end where m = \frac\pi R^3 \rho is the mass of the sphere.


Rigid body

If a
mechanical system A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...
is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis \mathbf 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 n particles, P_i, i = 1, \dots, n, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point \mathbf, and absolute velocities \mathbf_i: \begin \Delta\mathbf_i &= \mathbf_i - \mathbf, \\ \mathbf_i &= \boldsymbol \times \left(\mathbf_i - \mathbf\right) + \mathbf = \boldsymbol \times \Delta\mathbf_i + \mathbf, \end where \boldsymbol is the angular velocity of the system and \mathbf is the velocity of \mathbf. For planar movement the angular velocity vector is directed along the unit vector \mathbf which is perpendicular to the plane of movement. Introduce the unit vectors \mathbf_i from the reference point \mathbf to a point \mathbf_i, and the unit vector \mathbf_i = \mathbf \times \mathbf_i, so \begin \mathbf_i &= \frac,\quad \mathbf = \frac,\quad \mathbf_i = \mathbf \times \mathbf_i, \\ \mathbf_i &= \boldsymbol \times \Delta\mathbf_i + \mathbf = \omega\mathbf \times \Delta r_i\mathbf_i + \mathbf = \omega\, \Delta r_i\mathbf_i + \mathbf \end 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 \begin \mathbf &= \sum_^n m_i \Delta\mathbf_i \times \mathbf_i \\ &= \sum_^n m_i \,\Delta r_i\mathbf_i \times \left(\omega\, \Delta r_i\mathbf_i + \mathbf\right) \\ &= \left(\sum_^n m_i \,\Delta r_i^2\right)\omega \mathbf + \left(\sum_^n m_i\,\Delta r_i\mathbf_i\right) \times \mathbf. \end Use the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
\mathbf as the reference point so \begin \Delta r_i \mathbf_i &= \mathbf_i - \mathbf, \\ \sum_^n m_i\,\Delta r_i \mathbf_i &= 0, \end and define the moment of inertia relative to the center of mass I_\mathbf as I_\mathbf = \sum_ m_i\,\Delta r_i^2, then the equation for angular momentum simplifies to \mathbf = I_\mathbf \omega \mathbf. The moment of inertia I_\mathbf 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 \begin E_\text &= \frac \sum_^n m_i \mathbf_i \cdot \mathbf_i, \\ &= \frac \sum_^n m_i \left(\omega \,\Delta r_i\mathbf_i + \mathbf\right) \cdot \left(\omega \,\Delta r_i\mathbf_i + \mathbf\right), \\ &= \frac\omega^2 \left(\sum_^n m_i\, \Delta r_i^2 \mathbf_i \cdot \mathbf_i\right) + \omega\mathbf \cdot \left(\sum_^n m_i \,\Delta r_i\mathbf_i\right) + \frac\left(\sum_^n m_i\right) \mathbf \cdot \mathbf. \end Let the reference point be the center of mass \mathbf of the system so the second term becomes zero, and introduce the moment of inertia I_\mathbf so the kinetic energy is given by E_\text = \frac I_\mathbf \omega^2 + \frac M\mathbf \cdot \mathbf. The moment of inertia I_\mathbf is the ''polar moment of inertia'' of the body.


Newton's laws

Newton's laws for a rigid system of n particles, P_i, i = 1, \dots, n, can be written in terms of a
resultant force In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant for ...
and torque at a reference point \mathbf, to yield \begin \mathbf &= \sum_^n m_i\mathbf_i, \\ \boldsymbol\tau &= \sum_^n \Delta\mathbf_i \times m_i\mathbf_i, \end where \mathbf_i denotes the trajectory of each particle. The
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
of a rigid body yields the formula for the acceleration of the particle P_i in terms of the position \mathbf and acceleration \mathbf of the reference particle as well as the angular velocity vector \boldsymbol and angular acceleration vector \boldsymbol of the rigid system of particles as, \mathbf_i = \boldsymbol\alpha \times \Delta\mathbf_i + \boldsymbol \times \boldsymbol \times \Delta\mathbf_i + \mathbf. For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along \mathbf 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 \mathbf_i from the reference point \mathbf to a point \mathbf_i and the unit vectors \mathbf_i = \mathbf \times \mathbf_i, so \begin \mathbf_i &= \alpha\mathbf \times \Delta r_i\mathbf_i - \omega\mathbf \times \omega\mathbf \times \Delta r_i\mathbf_i + \mathbf \\ &= \alpha \Delta r_i\mathbf_i - \omega^2 \Delta r_i\mathbf_i + \mathbf. \end This yields the resultant torque on the system as \begin \boldsymbol &= \sum_^n m_i\,\Delta r_i\mathbf_i \times \left(\alpha\Delta r_i\mathbf_i - \omega^2\Delta r_i\mathbf_i + \mathbf\right) \\ &= \left(\sum_^n m_i\,\Delta r_i^2\right)\alpha \mathbf + \left(\sum_^n m_i\,\Delta r_i\mathbf_i\right) \times\mathbf, \end where \mathbf_i \times \mathbf_i = \mathbf, and \mathbf_i \times \mathbf_i = \mathbf is the unit vector perpendicular to the plane for all of the particles P_i. Use the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
\mathbf as the reference point and define the moment of inertia relative to the center of mass I_\mathbf, then the equation for the resultant torque simplifies to \boldsymbol = I_\mathbf\alpha\mathbf.


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.L. W. Tsai, Robot Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999. Let the system of n particles, P_i, i = 1, \dots, n be located at the coordinates \mathbf_i with velocities \mathbf_i relative to a fixed reference frame. For a (possibly moving) reference point \mathbf, the relative positions are \Delta\mathbf_i = \mathbf_i - \mathbf and the (absolute) velocities are \mathbf_i = \boldsymbol \times \Delta\mathbf_i + \mathbf_\mathbf where \boldsymbol is the angular velocity of the system, and \mathbf is the velocity of \mathbf.


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, \left mathbf\right/math>, constructed from the components of \mathbf = (b_x, b_y, b_z): \begin \mathbf \times \mathbf &\equiv \left mathbf\right\mathbf \\ \left mathbf\right&\equiv \begin 0 & -b_z & b_y \\ b_z & 0 & -b_x \\ -b_y & b_x & 0 \end. \end The inertia matrix is constructed by considering the angular momentum, with the reference point \mathbf of the body chosen to be the center of mass \mathbf: \begin \mathbf &= \sum_^n m_i\,\Delta\mathbf_i \times \mathbf_i \\ &= \sum_^n m_i\,\Delta\mathbf_i \times \left(\boldsymbol \times \Delta\mathbf_i + \mathbf_\mathbf\right) \\ &= \left(-\sum_^n m_i\,\Delta\mathbf_i \times \left(\Delta\mathbf_i \times \boldsymbol\right)\right) + \left(\sum_^n m_i \,\Delta\mathbf_i \times \mathbf_\mathbf\right), \end where the terms containing \mathbf (= \mathbf) sum to zero by the definition of
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. Then, the skew-symmetric matrix Delta\mathbf_i/math> obtained from the relative position vector \Delta\mathbf_i = \mathbf_i - \mathbf, can be used to define, \mathbf = \left(-\sum_^n m_i \left Delta\mathbf_i\right2\right)\boldsymbol = \mathbf_\mathbf \boldsymbol, where \mathbf defined by \mathbf_\mathbf = -\sum_^n m_i \left Delta\mathbf_i\right2, is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass \mathbf.


Kinetic energy

The kinetic energy of a rigid system of particles can be formulated in terms of the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
and a matrix of mass moments of inertia of the system. Let the system of n particles P_i, i = 1, \dots, n be located at the coordinates \mathbf_i with velocities \mathbf_i, then the kinetic energy is E_\text = \frac \sum_^n m_i \mathbf_i \cdot \mathbf_i = \frac \sum_^n m_i \left(\boldsymbol \times \Delta\mathbf_i + \mathbf_\mathbf\right) \cdot \left(\boldsymbol \times \Delta\mathbf_i + \mathbf_\mathbf\right), where \Delta\mathbf_i = \mathbf_i - \mathbf is the position vector of a particle relative to the center of mass. This equation expands to yield three terms E_\text = \frac\left(\sum_^n m_i \left(\boldsymbol \times \Delta\mathbf_i\right) \cdot \left(\boldsymbol \times \Delta\mathbf_i\right)\right) + \left(\sum_^n m_i \mathbf_\mathbf \cdot \left(\boldsymbol \times \Delta\mathbf_i\right)\right) + \frac\left(\sum_^n m_i \mathbf_\mathbf \cdot \mathbf_\mathbf\right). Since the center of mass is defined by \sum_^n m_i \Delta\mathbf_i =0 , the second term in this equation is zero. Introduce the skew-symmetric matrix Delta\mathbf_i/math> so the kinetic energy becomes \begin E_\text &= \frac\left(\sum_^n m_i \left(\left Delta\mathbf_i\right\boldsymbol\right) \cdot \left(\left Delta\mathbf_i\right\boldsymbol\right)\right) + \frac\left(\sum_^n m_i\right) \mathbf_\mathbf \cdot \mathbf_\mathbf \\ &= \frac\left(\sum_^n m_i \left(\boldsymbol^\mathsf\left Delta\mathbf_i\right\mathsf \left Delta\mathbf_i\right\boldsymbol\right)\right) + \frac\left(\sum_^n m_i\right) \mathbf_\mathbf \cdot \mathbf_\mathbf \\ &= \frac\boldsymbol \cdot \left(-\sum_^n m_i \left Delta\mathbf_i\right2\right) \boldsymbol + \frac\left(\sum_^n m_i\right) \mathbf_\mathbf \cdot \mathbf_\mathbf. \end Thus, the kinetic energy of the rigid system of particles is given by E_\text = \frac \boldsymbol \cdot \mathbf_\mathbf \boldsymbol + \frac M\mathbf_\mathbf^2. where \mathbf is the inertia matrix relative to the center of mass and M 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, \boldsymbol = \sum_^n \left(\mathbf - \mathbf\right) \times m_i\mathbf_i, where \mathbf_i is the acceleration of the particle P_i. The
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
of a rigid body yields the formula for the acceleration of the particle P_i in terms of the position \mathbf and acceleration \mathbf_\mathbf of the reference point, as well as the angular velocity vector \boldsymbol and angular acceleration vector \boldsymbol of the rigid system as, \mathbf_i = \boldsymbol \times \left(\mathbf_i - \mathbf\right) + \boldsymbol \times \left( \boldsymbol \times \left(\mathbf_i - \mathbf\right) \right) + \mathbf_\mathbf. Use the center of mass \mathbf as the reference point, and introduce the skew-symmetric matrix \left Delta\mathbf_i\right= \left mathbf_i - \mathbf\right/math> to represent the cross product (\mathbf_i - \mathbf) \times, to obtain \boldsymbol = \left(-\sum_^n m_i\left Delta\mathbf_i\right2\right)\boldsymbol + \boldsymbol \times \left(-\sum_^n m_i \left Delta\mathbf_i\right2\right)\boldsymbol The calculation uses the identity \Delta\mathbf_i \times \left(\boldsymbol \times \left(\boldsymbol \times \Delta\mathbf_i\right)\right) + \boldsymbol \times \left(\left(\boldsymbol \times \Delta\mathbf_i\right) \times \Delta\mathbf_i\right) = 0, obtained from the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
for the triple
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
as shown in the proof below: Thus, the resultant torque on the rigid system of particles is given by \boldsymbol = \mathbf_\mathbf \boldsymbol + \boldsymbol \times \mathbf_\mathbf \boldsymbol, where \mathbf 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 \mathbf and the inertia matrix relative to another point \mathbf. This relationship is called the parallel axis theorem. Consider the inertia matrix \mathbf obtained for a rigid system of particles measured relative to a reference point \mathbf, given by \mathbf_\mathbf = -\sum_^n m_i\left mathbf_i - \mathbf\right2. Let \mathbf be the center of mass of the rigid system, then \mathbf = (\mathbf - \mathbf) + \mathbf = \mathbf + \mathbf, where \mathbf is the vector from the center of mass \mathbf to the reference point \mathbf. Use this equation to compute the inertia matrix, \mathbf_\mathbf = -\sum_^n m_i mathbf_i - \left(\mathbf + \mathbf\right)2 = -\sum_^n m_i left(\mathbf_i - \mathbf\right) - \mathbf2. Distribute over the cross product to obtain \mathbf_\mathbf = - \left(\sum_^n m_i mathbf_i - \mathbf2\right) + \left(\sum_^n m_i mathbf_i - \mathbfright) mathbf + mathbfleft(\sum_^n m_i mathbf_i - \mathbfright) - \left(\sum_^n m_i\right) mathbf2. The first term is the inertia matrix \mathbf relative to the center of mass. The second and third terms are zero by definition of the center of mass \mathbf. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix mathbf/math> constructed from \mathbf. The result is the parallel axis theorem, \mathbf_\mathbf = \mathbf_\mathbf - M mathbf2, where \mathbf is the vector from the center of mass \mathbf to the reference point \mathbf. 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 -m\left mathbf\right2, which is similar to the mr^2 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 m\left mathbf\right\mathsf \left mathbf\right/math>, if desired, by using the skew-symmetry property of mathbf/math>.


Scalar moment of inertia in a plane

The scalar moment of inertia, I_L, of a body about a specified axis whose direction is specified by the unit vector \mathbf and passes through the body at a point \mathbf is as follows: I_L = \mathbf \cdot \left(-\sum_^N m_i \left Delta\mathbf_i\right2 \right) \mathbf = \mathbf \cdot \mathbf_\mathbf \mathbf = \mathbf^\mathsf \mathbf_\mathbf \mathbf, where \mathbf is the moment of inertia matrix of the system relative to the reference point \mathbf, and Delta\mathbf_i/math> is the skew symmetric matrix obtained from the vector \Delta\mathbf_i = \mathbf_i - \mathbf. This is derived as follows. Let a rigid assembly of n particles, P_i, i = 1, \dots, n, have coordinates \mathbf_i. Choose \mathbf as a reference point and compute the moment of inertia around a line L defined by the unit vector \mathbf through the reference point \mathbf, \mathbf(t) = \mathbf + t\mathbf. The perpendicular vector from this line to the particle P_i is obtained from \Delta\mathbf_i by removing the component that projects onto \mathbf. \Delta\mathbf_i^\perp = \Delta\mathbf_i - \left(\mathbf \cdot \Delta\mathbf_i\right)\mathbf = \left(\mathbf - \mathbf\mathbf^\mathsf\right) \Delta\mathbf_i, where \mathbf is the identity matrix, so as to avoid confusion with the inertia matrix, and \mathbf\mathbf^\mathsf is the outer product matrix formed from the unit vector \mathbf along the line L. To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix \left mathbf\right/math> such that \left mathbf\rightmathbf = \mathbf \times \mathbf, then we have the identity -\left mathbf\right2 \equiv \left, \mathbf\^2\left(\mathbf - \mathbf\mathbf^\mathsf\right) = \mathbf - \mathbf\mathbf^\mathsf, noting that \mathbf is a unit vector. The magnitude squared of the perpendicular vector is \begin \left, \Delta\mathbf_i^\perp\^2 &= \left(-\left mathbf\right2 \Delta\mathbf_i\right) \cdot \left(-\left mathbf\right2 \Delta\mathbf_i\right) \\ &= \left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \cdot \left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \end The simplification of this equation uses the triple scalar product identity \left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \cdot \left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \equiv \left(\left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \times \mathbf\right) \cdot \left(\mathbf \times \Delta\mathbf_i\right), where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting that \Delta\mathbf_i and \mathbf are orthogonal: \begin &\left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \cdot \left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \\ = &\left(\left(\mathbf \times \left(\mathbf \times \Delta\mathbf_i\right)\right) \times \mathbf\right) \cdot \left(\mathbf \times \Delta\mathbf_i\right) \\ = &\left(\mathbf \times \Delta\mathbf_i\right) \cdot \left(-\Delta\mathbf_i \times \mathbf\right) \\ = &-\mathbf \cdot \left(\Delta\mathbf_i \times \Delta\mathbf_i \times \mathbf\right) \\ = &-\mathbf \cdot \left Delta\mathbf_i\right2 \mathbf. \end Thus, the moment of inertia around the line L through \mathbf in the direction \mathbf is obtained from the calculation \begin I_L &= \sum_^N m_i \left, \Delta\mathbf_i^\perp\^2 \\ &= -\sum_^N m_i \mathbf \cdot \left Delta\mathbf_i\right2\mathbf = \mathbf \cdot \left(-\sum_^N m_i \left Delta\mathbf_i\right2 \right) \mathbf \\ &= \mathbf \cdot \mathbf_\mathbf \mathbf = \mathbf^\mathsf \mathbf_\mathbf \mathbf, \end where \mathbf is the moment of inertia matrix of the system relative to the reference point \mathbf. 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
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 N point masses m_, the moment of inertia
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
is given by \mathbf = \begin I_ & I_ & I_ \\ I_ & I_ & I_ \\ I_ & I_ & I_ \end. Its components are defined as I_ \ \stackrel\ \sum_^ m_\left(\left\, \mathbf_k\right\, ^\delta_ - x_^x_^\right) where * i, j is equal to 1, 2 or 3 for x, y, and z, respectively, * \mathbf_k = \left(x_1^, x_2^, x_3^\right) is the vector to the point mass m_k from the point about which the tensor is calculated and * \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. Note that, by the definition, \mathbf is a
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
. The diagonal elements are more succinctly written as \begin I_ \ &\stackrel\ \sum_^ m_ \left(y_^ + z_^\right), \\ I_ \ &\stackrel\ \sum_^ m_ \left(x_^ + z_^\right), \\ I_ \ &\stackrel\ \sum_^ m_ \left(x_^ + y_^\right), \end while the off-diagonal elements, also called the products of inertia, are \begin I_ = I_ \ &\stackrel\ -\sum_^ m_ x_ y_, \\ I_ = I_ \ &\stackrel\ -\sum_^ m_ x_ z_, \\ I_ = I_ \ &\stackrel\ -\sum_^ m_ y_ z_. \end Here I_ denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, I_ 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 distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has \mathbf = \iiint_V \rho(x,y,z) \left( \, \mathbf\, ^2 \mathbf_ - \mathbf\otimes \mathbf\right)\, dx \, dy \, dz, where \mathbf\otimes \mathbf is their
outer product In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
, E3 is the 3×3
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, and ''V'' is a region of space completely containing the object. Alternatively it can also be written in terms of the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
mathbf rmathbf x = \mathbf r\times\mathbf x: \mathbf = \iiint_V \rho(\mathbf) mathbf r\textsf mathbf r\, dV = -\iiint_ \rho(\mathbf) mathbf r2 \, dV 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 \mathbf, I_n = \mathbf\cdot\mathbf\cdot\mathbf, where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as I_ is obtained by the computation I_ = \mathbf_1\cdot\mathbf\cdot\mathbf_2, and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis. The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by, \mathbf = \begin I_ & I_ & I_ \\ I_ & I_ & I_ \\ I_ & I_ & I_ \end = \begin I_ & I_ & I_ \\ I_ & I_ & I_ \\ I_ & I_ & I_ \end = \begin \sum_^ m_ \left(y_^ + z_^\right) & -\sum_^ m_ x_ y_ & -\sum_^ m_ x_ z_ \\ -\sum_^ m_ x_ y_ & \sum_^ m_ \left(x_^ + z_^\right) & -\sum_^ m_ y_ z_ \\ -\sum_^ m_ x_ z_ & -\sum_^ m_ y_ z_ & \sum_^ m_ \left(x_^ + y_^\right) \end. It is common in rigid body mechanics to use notation that explicitly identifies the x, y, and z-axes, such as I_ and I_, 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: \begin I_ = I_ \ &\stackrel\ \sum_^ m_ x_ y_, \\ I_ = I_ \ &\stackrel\ \sum_^ m_ x_ z_, \\ I_ = I_ \ &\stackrel\ \sum_^ m_ y_ z_, \\ pt \mathbf = \begin I_ & I_ & I_ \\ I_ & I_ & I_ \\ I_ & I_ & I_ \end &= \begin I_ & -I_ & -I_ \\ -I_ & I_ & -I_ \\ -I_ & -I_ & I_ \end = \begin \sum_^ m_ \left(y_^ + z_^\right) & -\sum_^ m_ x_ y_ & -\sum_^ m_ x_ z_ \\ -\sum_^ m_ x_ y_ & \sum_^ m_ \left(x_^ + z_^\right) & -\sum_^ m_ y_ z_ \\ -\sum_^ m_ x_ z_ & -\sum_^ m_ y_ z_ & \sum_^ m_ \left(x_^ + y_^\right) \end. \end


Determine inertia convention (Principal axes method)

If one has the inertia data (I_, I_, I_, I_, I_, I_) without knowing which inertia convention that has been used, it can be determined if one also has the principal axes. With the principal axes method, one makes inertia matrices from the following two assumptions: # The standard inertia convention has been used (I_ = I_, I_ = I_, I_ = I_). # The alternate inertia convention has been used (I_ = -I_, I_ = -I_, I_ = -I_). Next, one calculates the eigenvectors for the two matrices. The matrix whose eigenvectors are parallel to the principal axes corresponds to the inertia convention that has been used.


Derivation of the tensor components

The distance r of a particle at \mathbf from the axis of rotation passing through the origin in the \mathbf direction is \left, \mathbf - \left(\mathbf \cdot \mathbf\right) \mathbf\, where \mathbf is unit vector. The moment of inertia on the axis is I = mr^2 = m\left(\mathbf - \left(\mathbf\cdot\mathbf\right) \mathbf\right)\cdot\left(\mathbf - \left(\mathbf\cdot\mathbf\right) \mathbf\right) = m\left(\mathbf^2 - 2\mathbf\left(\mathbf\cdot\mathbf\right)\mathbf + \left(\mathbf\cdot\mathbf\right)^2\mathbf^2\right) = m\left(\mathbf^2 - \left(\mathbf\cdot\mathbf\right)^2\right). Rewrite the equation using
matrix transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
: I = m\left(\mathbf^\textsf\mathbf - \mathbf^\textsf\mathbf\mathbf^\textsf\mathbf\right) = m\cdot\mathbf^\textsf\left(\mathbf^\textsf\mathbf\cdot\mathbf - \mathbf\mathbf^\textsf\right)\mathbf, where E3 is the 3×3
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
. This leads to a tensor formula for the moment of inertia I = m \begin n_1 & n_2 & n_3 \end\begin y^2 + z^2 & -xy & -xz \\ -yx & x^2 + z^2 & -yz \\ -zx & -zy & x^2 + y^2 \end \begin n_1 \\ n_2 \\ n_3 \end. 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 \mathbf_0 be the inertia tensor of a body calculated at its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, and \mathbf be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by: \mathbf = \mathbf_0 + m \mathbf\cdot\mathbf)\mathbf_3 - \mathbf\otimes\mathbf/math> where m is the body's mass, E3 is the 3 × 3 identity matrix, and \otimes is the
outer product In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
.


Inertia tensor of rotation

Let \mathbf be the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
that represents a body's rotation. The inertial tensor of the rotated body is given by: \mathbf = \mathbf\mathbf\mathbf^\textsf


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 \mathbf_\mathbf^B, and define the orientation of the body frame relative to the inertial frame by the rotation matrix \mathbf, such that, \mathbf = \mathbf\mathbf, where vectors \mathbf in the body fixed coordinate frame have coordinates \mathbf in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by \mathbf_\mathbf = \mathbf \mathbf_\mathbf^B \mathbf^\mathsf. Notice that \mathbf changes as the body moves, while \mathbf_\mathbf^B 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 In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...
into the product of a rotation matrix \mathbf and a diagonal matrix \boldsymbol, given by \mathbf_\mathbf^B = \mathbf\boldsymbol\mathbf^\mathsf, where \boldsymbol = \begin I_1 & 0 & 0 \\ 0 & I_2 & 0 \\ 0 & 0 & I_3 \end. The columns of the rotation matrix \mathbf define the directions of the principal axes of the body, and the constants I_1, I_2, and I_3 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 Sylvester'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 basis. Namely, if ''A'' is the symmetric matrix that defines the quadra ...
. The principal axis with the highest moment of inertia is sometimes called the figure axis or axis of figure. A toy
top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few ...
is an example of a rotating rigid body, and the word ''top'' is used in names the types of types of rigid bodies. When all principal moments of inertia are distinct, the principal axes through
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
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 m, meaning it is symmetrical under rotations of about the given axis, that axis is a principal axis. When m > 2, 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 In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
. The
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
of
vehicle A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor vehicles (motorcycles, cars, trucks, buses, mobility scooters for disabled people), railed vehicles (trains, trams), wa ...
s is often described in terms of
yaw, pitch, and roll An aircraft in flight is free to rotate in three dimensions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from ...
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 In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion ha ...
. Let \boldsymbol be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface \mathbf^\mathsf\boldsymbol\mathbf = 1, or I_1x^2 + I_2y^2 + I_3z^2 =1, defines an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
in the body frame. Write this equation in the form, \left(\frac\right)^2 + \left(\frac\right)^2 + \left(\frac\right)^2 = 1, to see that the semi-principal diameters of this ellipsoid are given by a = \frac, \quad b=\frac, \quad c=\frac. Let a point \mathbf on this ellipsoid be defined in terms of its magnitude and direction, \mathbf = \, \mathbf\, \mathbf, where \mathbf is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia I_\mathbf around an axis in the direction \mathbf, yields \mathbf^\mathsf\boldsymbol\mathbf = \, \mathbf\, ^2\mathbf^\mathsf\boldsymbol\mathbf = \, \mathbf\, ^2 I_\mathbf = 1. Thus, the magnitude of a point \mathbf in the direction \mathbf on the inertia ellipsoid is \, \mathbf\, = \frac.


See also

*
Central moment In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
*
List of moments of inertia Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
*
Planar lamina In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integration. Planar laminas can be use ...
*
Rotational energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the o ...
* Moment of inertia factor


References


External links


Angular momentum and rigid-body rotation in two and three dimensions






* ttp://hypertextbook.com/physics/mechanics/rotational-inertia/ Tutorial on finding moments of inertia, with problems and solutions on various basic shapes
Notes on mechanics of manipulation: the angular inertia tensor
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Easy to use and Free Moment of Inertia Calculator online