# Moment of inertia

# Moment of inertia

Moment of inertia | |
---|---|

Common symbols | I |

SI unit | kg m^{2} |

Other units | lbf·ft·s^{2} |

Extensive? | yes |

Derivations from other quantities | |

Dimension | M L^{2} |

Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the Niagara River in 1890.

The **moment of inertia**, otherwise known as the **angular mass** or **rotational inertia**, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate. It is an extensive (additive) property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

Moment of inertia | |
---|---|

Common symbols | I |

SI unit | kg m^{2} |

Other units | lbf·ft·s^{2} |

Extensive? | yes |

Derivations from other quantities | |

Dimension | M L^{2} |

Introduction

When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram meter squared (kg·m2) in SI units and pound-foot-second squared (lbf·ft·s2) in imperial or US units.

`Moment of inertia plays the role in rotational kinetics thatmass(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, whereis the distance of the point from the axis, andis 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 question. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.`

In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum.^{[1]} The term *moment of inertia* was introduced by Leonhard Euler in his book *Theoria motus corporum solidorum seu rigidorum* in 1765,^{[1]}^{[2]} 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.^{[3]}^{[4]}

Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.^{[5]}^{[6]}

The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axis determines how steering forces on the control surfaces of its wings, elevators and tail affect the plane in roll, pitch and yaw.

Definition

Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.

Video of rotating chair experiment, illustrating moment of inertia. When the spinning professor pulls his arms, his moment of inertia decreases; to conserve angular momentum, his angular velocity increases.

`Moment of inertiais defined as the ratio of the netangular momentumof a system to itsangular velocityaround a principal axis,`

^{[7]}^{[8]}that isIf the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster.^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}

`If the shape of the body does not change, then its moment of inertia appears inNewton's law of motionas the ratio of anapplied torqueon a body to theangular accelerationaround a principal axis, that is`

`For asimple pendulum, this definition yields a formula for the moment of inertiain terms of the massof the pendulum and its distancefrom the pivot point as,`

`Thus, moment of inertia depends on both the massof a body and its geometry, or shape, as defined by the distanceto 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 masseseach multiplied by the square of its perpendicular distanceto an axis.`

`In general, given an object of mass, an effective radiuscan be defined for an axis through its center of mass, with such a value that its moment of inertia is`

`whereis known as theradius of gyration.`

Examples

Simple pendulum

`Moment of inertia can be measured using a simple pendulum, because it is the resistance to the rotation caused by gravity. Mathematically, the moment of inertia of the pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particlewith the square of its distanceto the pivot, that is`

`This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torquearound the axis perpendicular to the plane of the pendulum movement. Hereis the distance vector perpendicular to and from the force to the torque axis, andis the net force on the mass. Associated with this torque is anangular acceleration,, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is. Sincethe torque equation becomes:`

`whereis a unit vector perpendicular to the plane of the pendulum. (The second to last step uses thevector triple product expansionwith the perpendicularity ofand.) The quantityis the`

*moment of inertia*of this single mass around the pivot point.`The quantityalso appears in theangular momentumof a simple pendulum, which is calculated from the velocityof the pendulum mass around the pivot, whereis theangular velocityof the mass about the pivot point. This angular momentum is given by`

using a similar derivation to the previous equation.

Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield

`This shows that the quantityis 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 valuesfor all of the elements of mass in the body.`

Compound pendulum

Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.

`Acompound pendulumis 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.`

^{[14]}^{[15]}:395–396^{[16]}:51–53Thenaturalfrequency() of a compound pendulum depends on its moment of inertia,,`whereis the mass of the object,is local acceleration of gravity, andis 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.`

^{[17]}:516–517`Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot pointso 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 (), to obtain`

`whereis the period (duration) of oscillation (usually averaged over multiple periods).`

`The moment of inertia of the body about itscenter of mass,, is then calculated using theparallel axis theoremto be`

`whereis the mass of the body andis the distance from the pivot pointto the center of mass.`

`Moment of inertia of a body is often defined in terms of its *radius of gyration*, which is the radius of a ring of equal mass around the center of mass of a body that has the same moment of inertia. The radius of gyrationis calculated from the body's moment of inertiaand massas the length`

^{[18]}:1296–1297Center of oscillation

`A simple pendulum that has the same natural frequency as a compound pendulum defines the lengthfrom the pivot to a point called thecenter of oscillationof the compound pendulum. This point also corresponds to thecenter of percussion. The lengthis determined from the formula,`

or

`Theseconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency offor the pendulum. In this case, the distance to the center of oscillation,, can be computed to be`

Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.

Measuring moment of inertia

The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.^{[19]} The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.^{[20]}

Motion in a fixed plane

Point mass

Four objects with identical masses and radii racing down a plane while rolling without slipping. From back to front: spherical shell, solid sphere, cylindrical ring, and solid cylinder. The time for each object to reach the finishing line depends on their moment of inertia. (OGV version)

`The moment of inertia about an axis of a body is calculated by summingfor every particle in the body, whereis 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.`

^{[21]})`Consider the kinetic energy of an assembly ofmassesthat lie at the distancesfrom the pivot point, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,`

^{[17]}:516–517^{[18]}:1084–1085^{[18]}:1296–1300`This shows that the moment of inertia of the body is the sum of each of theterms, that is`

Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.

The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:

Another expression replaces the summation with an integral,

`Here, the functiongives the mass density at each point,is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a pointin the solid, and the integration is evaluated over the volumeof the body. 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 thesecond moment of areaof 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-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- and

^{[22]}The stresses in abeamare calculated using the second moment of the cross-sectional area around either the-axis or-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.^{[18]}

A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly.

As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that form the sphere. If the surface of the ball is defined by the equation^{[18]} ^{[]}

`then the radiusof the disc at the cross-sectionalong the-axis is`

`Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the-axis,`

`whereis the mass of the sphere.`

Rigid body

`If amechanical systemis constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axisperpendicular 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.^{[14]}^{[17]}^{[23]}^{[24]}`If a system ofparticles,, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point, and absolute velocities:`

`whereis the angular velocity of the system andis the velocity of.`

`For planar movement the angular velocity vector is directed along the unit vectorwhich is perpendicular to the plane of movement. Introduce the unit vectorsfrom the reference pointto a point, and the unit vector, so`

This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.

**Note on the cross product**: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.

Angular momentum

The angular momentum vector for the planar movement of a rigid system of particles is given by^{[14]}^{[17]}

`Use thecenter of massas the reference point so`

`and define the moment of inertia relative to the center of massas`

then the equation for angular momentum simplifies to^{[18]} ^{[]}

`The moment of inertiaabout 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 thesecond moment of masswith 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

This 1906 rotary shear uses the moment of inertia of two flywheels to store kinetic energy which when released is used to cut metal stock (International Library of Technology, 1906).

The kinetic energy of a rigid system of particles moving in the plane is given by^{[14]}^{[17]}

`Let the reference point be the center of massof the system so the second term becomes zero, and introduce the moment of inertiaso the kinetic energy is given by`

^{[18]}:1084`The moment of inertiais the`

*polar moment of inertia*of the body.Newton's laws

A 1920s John Deere tractor with the spoked flywheel on the engine. The large moment of inertia of the flywheel smooths the operation of the tractor

`Newton's laws for a rigid system ofparticles,, can be written in terms of aresultant forceand torque at a reference point, to yield`

^{[14]}^{[17]}`wheredenotes the trajectory of each particle.`

`Thekinematicsof a rigid body yields the formula for the acceleration of the particlein terms of the positionand accelerationof the reference particle as well as the angular velocity vectorand angular acceleration vectorof the rigid system of particles as,`

`For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed alongperpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectorsfrom the reference pointto a pointand the unit vectors, so`

This yields the resultant torque on the system as

`where, andis the unit vector perpendicular to the plane for all of the particles.`

`Use thecenter of massas the reference point and define the moment of inertia relative to the center of mass, then the equation for the resultant torque simplifies to`

^{[18]}:1029Motion 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.^{[3]}^{[4]}^{[5]}^{[6]}^{[25]}

`Let the system ofparticles,be located at the coordinateswith velocitiesrelative to a fixed reference frame. For a (possibly moving) reference point, the relative positions are`

and the (absolute) velocities are

`whereis the angular velocity of the system, andis the velocity of.`

Angular momentum

`Note that thecross product can be equivalently written as matrix multiplicationby combining the first operand and the operator into a, skew-symmetric, matrix,, constructed from the components of:`

`The inertia matrix is constructed by considering the angular momentum, with the reference pointof the body chosen to be the center of mass:`

^{[3]}^{[6]}`where the terms containing() sum to zero by the definition ofcenter of mass.`

`Then, the skew-symmetric matrixobtained from the relative position vector, can be used to define,`

`wheredefined by`

`is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass.`

Kinetic energy

`The kinetic energy of a rigid system of particles can be formulated in terms of thecenter of massand a matrix of mass moments of inertia of the system. Let the system ofparticlesbe located at the coordinateswith velocities, then the kinetic energy is`

^{[3]}^{[6]}`whereis the position vector of a particle relative to the center of mass.`

This equation expands to yield three terms

`The second term in this equation is zero becauseis the center of mass. Introduce the skew-symmetric matrixso the kinetic energy becomes`

Thus, the kinetic energy of the rigid system of particles is given by

`whereis the inertia matrix relative to the center of mass andis 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,^{[3]}^{[6]}

`whereis the acceleration of the particle. Thekinematicsof a rigid body yields the formula for the acceleration of the particlein terms of the positionand accelerationof the reference point, as well as the angular velocity vectorand angular acceleration vectorof the rigid system as,`

`Use the center of massas the reference point, and introduce the skew-symmetric matrixto represent the cross product, to obtain`

The calculation uses the identity

obtained from the Jacobi identity for the triple cross product as shown in the proof below:

Proof |
---|

Then, the followingJacobi identityis used on the last term: |

Thus, the resultant torque on the rigid system of particles is given by

`whereis 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 massand the inertia matrix relative to another point. This relationship is called the parallel axis theorem.`

^{[3]}^{[6]}`Consider the inertia matrixobtained for a rigid system of particles measured relative to a reference point, given by`

`Letbe the center of mass of the rigid system, then`

`whereis the vector from the center of massto the reference point. Use this equation to compute the inertia matrix,`

Distribute over the cross product to obtain

`The first term is the inertia matrixrelative to the center of mass. The second and third terms are zero by definition of the center of mass. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrixconstructed from.`

The result is the parallel axis theorem,

`whereis the vector from the center of massto the reference point.`

**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, which is similar to thethat 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, if desired, by using the skew-symmetry property of.Scalar moment of inertia in a plane

`The scalar moment of inertia,, of a body about a specified axis whose direction is specified by the unit vectorand passes through the body at a pointis as follows:`

^{[6]}`whereis the moment of inertia matrix of the system relative to the reference point, andis the skew symmetric matrix obtained from the vector.`

`This is derived as follows. Let a rigid assembly ofparticles,, have coordinates. Chooseas a reference point and compute the moment of inertia around a line L defined by the unit vectorthrough the reference point,. The perpendicular vector from this line to the particleis obtained fromby removing the component that projects onto.`

`whereis the identity matrix, so as to avoid confusion with the inertia matrix, andis the outer product matrix formed from the unit vectoralong the line.`

`To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrixsuch that, then we have the identity`

`noting thatis a unit vector.`

The magnitude squared of the perpendicular vector is

The simplification of this equation uses the triple scalar product identity

`where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting thatandare orthogonal:`

`Thus, the moment of inertia around the linethroughin the directionis obtained from the calculation`

`whereis the moment of inertia matrix of the system relative to the reference point.`

This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.

Inertia tensor

For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The **moment of inertia tensor** is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.

Definition

`For a rigid object ofpoint masses, the moment of inertiatensoris given by`

- .

Its components are defined as

where

- is the

*i*,

*j*equal 1, 2, or 3 for x, y, and z, respectively,

**r**=(

*x*

_{1},

*x*

_{2},

*x*

_{3}) is the vector to the mass element

*dm*from the point about which the tensor is calculated,

*r*=

*||x||*, and

Note that, by the definition, *I* is a symmetric tensor.

The diagonal elements, also called the **principal moments of inertia**, are more succinctly written as

while the off-diagonal elements, also called the **products of inertia**, are

- and

`Heredenotes the moment of inertia around the-axis when the objects are rotated around the x-axis,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

`whereis theirouter product,`

**E**3is the 3 × 3identity matrix, and*V*is a region of space completely containing the object.`Alternatively it can also be written in terms of theangular momentum operator:`

`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,`

`where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such asis obtained by the computation`

`and can be interpreted as the moment of inertia around the-axis when the object rotates around the-axis.`

The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,

`It is common in rigid body mechanics to use notation that explicitly identifies the,, and-axes, such asand, for the components of the inertia tensor.`

Derivation of the tensor components

`The distanceof a particle atfrom the axis of rotation passing through the origin in thedirection 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 thedirection) isThis is aquadratic forminand, after a bit more algebra, this leads to a tensor formula for the moment of inertia`

- .

For multiple particles we need only recall that the moment of inertia is additive in order to see that this formula is correct.

Inertia 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.^{[6]}^{[23]} 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, and define the orientation of the body frame relative to the inertial frame by the rotation matrix, such that,`

`where vectorsin the body fixed coordinate frame have coordinatesin the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by`

`Notice thatchanges as the body moves, whileremains constant.`

Principal axes

`Measured in the body frame the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has theeigendecompositioninto the product of a rotation matrixand a diagonal matrix, given by`

where

`The columns of the rotation matrixdefine the directions of the principal axes of the body, and the constants,, andare called the`

**principal moments of inertia**. This result was first shown byJ. J. Sylvester (1852), and is a form ofSylvester's law of inertia.^{[26]}^{[27]}The principal axis with the highest moment of inertia is sometimes called the**figure axis**or**axis of figure**.When all principal moments of inertia are distinct, the principal axes through center of mass are uniquely specified. 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, meaning it is symmetrical under rotations of360°/`

*m*about the given axis, that 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, for example, a cube or any otherPlatonic solid.The motion of vehicles is often described in terms of yaw, pitch, and roll which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.

A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.

Ellipsoid

An ellipsoid with the semi-principal diameters labelled , , and .

`The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body calledPoinsot's ellipsoid.`

^{[28]}Letbe the inertia matrix relative to the center of mass aligned with the principal axes, then the surfaceor

defines an ellipsoid in the body frame. Write this equation in the form,

to see that the semi-principal diameters of this ellipsoid are given by

`Let a pointon this ellipsoid be defined in terms of its magnitude and direction,, whereis a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertiaaround an axis in the direction, yields`

`Thus, the magnitude of a pointin the directionon the inertia ellipsoid is`

See also

Central moment

List of moments of inertia

Rotational energy

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