moment of inertia basics understanding

8/13/2019 Moment of Inertia Basics Understanding

1/20

Moment of inertia

A tightrope walker uses the moment of inertia of the long rod to help maintain balance. This is Samuel Dixon

crossing the Niagara river in 18!.

Moment of inertiais the mass propert" of a rigidbod"that defines the tor#ue needed for a desired change inangular velocit"about an axis of rotation. $oment of inertia depends on the shape of the bod" and ma" be

different around different axes of rotation. A larger moment of inertia around a given axis re#uires more tor#ue

to increase the rotation% or to stop the rotation% of a bod" about that axis. $oment of inertia depends on the

amount and distribution of its mass% and can be found through the sum of moments of inertia of the masses

making up the whole ob&ect% under the same conditions. 'or example% if % then

. (n classical mechanics% moment of inertiama" also be called mass moment of inertia%

rotational inertia% polar moment of inertia% or the angular mass.

'or planar movement of a bod"% the tra&ectories of all of its points lie in parallel planes% and the rotation occurs

onl" about an axis perpendicular to this plane. (n this case% the bod" has a single moment of inertia% which is

measured around this axis.

'or spatial movement of a bod"% the moment of inertia is defined b" its s"mmetric )x) inertia matrix. Theinertia matrix is often described as a s"mmetricranktwo tensor%having six independent components. The

inertia matrix includes off*diagonal terms called products of inertia that couple tor#ue around one axis to

acceleration about another axis. +ach bod" has a set of mutuall" perpendicular axes% calledprincipal axes% forwhich the off*diagonal terms of the inertia matrix are ,ero% and a tor#ue around a principal axis onl" affects the

acceleration about that axis.

Contents

1 (ntroduction

o 1.1 Simple pendulum

o 1.- ompound pendulum

o 1.) enter of oscillation

- $easuring moment of inertia

) Definition

/ alculating moment of inertia about an axis

o /.1 +xample calculation of moment of inertia

0 $oment of inertia in planar movement of a rigid bod"
http://en.wikipedia.org/wiki/Physical_bodyhttp://en.wikipedia.org/wiki/Physical_bodyhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Principal_axis_(mechanics)http://en.wikipedia.org/wiki/Moment_of_inertia#Introductionhttp://en.wikipedia.org/wiki/Moment_of_inertia#Simple_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Center_of_oscillationhttp://en.wikipedia.org/wiki/Moment_of_inertia#Measuring_moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#Definitionhttp://en.wikipedia.org/wiki/Moment_of_inertia#Calculating_moment_of_inertia_about_an_axishttp://en.wikipedia.org/wiki/Moment_of_inertia#Example_calculation_of_moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#Moment_of_inertia_in_planar_movement_of_a_rigid_bodyhttp://en.wikipedia.org/wiki/File:Samuel_Dixon_Niagara.jpghttp://en.wikipedia.org/wiki/File:Samuel_Dixon_Niagara.jpghttp://en.wikipedia.org/wiki/Physical_bodyhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Principal_axis_(mechanics)http://en.wikipedia.org/wiki/Moment_of_inertia#Introductionhttp://en.wikipedia.org/wiki/Moment_of_inertia#Simple_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Center_of_oscillationhttp://en.wikipedia.org/wiki/Moment_of_inertia#Measuring_moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#Definitionhttp://en.wikipedia.org/wiki/Moment_of_inertia#Calculating_moment_of_inertia_about_an_axishttp://en.wikipedia.org/wiki/Moment_of_inertia#Example_calculation_of_moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#Moment_of_inertia_in_planar_movement_of_a_rigid_body


2/20

o 0.1 Angular momentum in planar movement

o 0.- inetic energ" in planar movement

o 0.) Newton2s laws for planar movement

3 The inertia matrix for spatial movement of a rigid bod"

o 3.1 Angular momentum

o 3.- inetic energ"

o 3.) 4esultant tor#ue

o 3./ 5arallel axis theorem

6 The inertia matrix and the scalar moment of inertia around an arbitrar" axis 8 The inertia tensor

o 8.1 (dentities for a skew*s"mmetric matrix

The inertia matrix in different reference frames

o .1 7od" frame inertia matrix

o .- 5rincipal axes

o .) (nertia ellipsoid

1! See also

11 4eferences

1- +xternal links

Introduction

A fl"wheelis a wheel with a large moment of inertia used to smooth out motion in machines. This example is ina 4ussian museum.

hen a bod" is rotating around an axis% ator#uemust be applied to change its angular momentum. The amount

of tor#ue needed for an" given change in angular momentum is proportional to the si,e of that change. $oment

of inertia has units of kg9m-in S(units and lbmft-: in;Sunits.

(n 136) hristiaan =->and it is incorporated into +uler2s secondlaw.

The natural fre#uenc" of oscillation of a compound pendulum is obtained from the ratio of the tor#ue imposed

b" gravit" on the mass of the pendulum to the resistance to acceleration defined b" the moment of inertia.

omparison of this natural fre#uenc" to that of a simple pendulum consisting of a single point of mass providesa mathematical formulation for moment of inertia of an extended bod".=)>=/>
http://en.wikipedia.org/wiki/Moment_of_inertia#Angular_momentum_in_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#Kinetic_energy_in_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#Newton.27s_laws_for_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_for_spatial_movement_of_a_rigid_bodyhttp://en.wikipedia.org/wiki/Moment_of_inertia#Angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Kinetic_energyhttp://en.wikipedia.org/wiki/Moment_of_inertia#Resultant_torquehttp://en.wikipedia.org/wiki/Moment_of_inertia#Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_and_the_scalar_moment_of_inertia_around_an_arbitrary_axishttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_tensorhttp://en.wikipedia.org/wiki/Moment_of_inertia#Identities_for_a_skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_in_different_reference_frameshttp://en.wikipedia.org/wiki/Moment_of_inertia#Body_frame_inertia_matrixhttp://en.wikipedia.org/wiki/Moment_of_inertia#Principal_axeshttp://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_ellipsoidhttp://en.wikipedia.org/wiki/Moment_of_inertia#See_alsohttp://en.wikipedia.org/wiki/Moment_of_inertia#Referenceshttp://en.wikipedia.org/wiki/Moment_of_inertia#External_linkshttp://en.wikipedia.org/wiki/Flywheelhttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/United_States_customary_unitshttp://en.wikipedia.org/wiki/United_States_customary_unitshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-mach-1http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-mach-1http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Euler1730-2http://en.wikipedia.org/wiki/Euler's_laws#Euler.27s_second_lawhttp://en.wikipedia.org/wiki/Euler's_laws#Euler.27s_second_lawhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Symon_1971-4http://en.wikipedia.org/wiki/File:%D0%9C%D0%B0%D1%85%D0%BE%D0%B2%D0%B8%D0%BA.jpghttp://en.wikipedia.org/wiki/File:%D0%9C%D0%B0%D1%85%D0%BE%D0%B2%D0%B8%D0%BA.jpghttp://en.wikipedia.org/wiki/Moment_of_inertia#Angular_momentum_in_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#Kinetic_energy_in_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#Newton.27s_laws_for_planar_movementhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_for_spatial_movement_of_a_rigid_bodyhttp://en.wikipedia.org/wiki/Moment_of_inertia#Angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertia#Kinetic_energyhttp://en.wikipedia.org/wiki/Moment_of_inertia#Resultant_torquehttp://en.wikipedia.org/wiki/Moment_of_inertia#Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_and_the_scalar_moment_of_inertia_around_an_arbitrary_axishttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_tensorhttp://en.wikipedia.org/wiki/Moment_of_inertia#Identities_for_a_skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Moment_of_inertia#The_inertia_matrix_in_different_reference_frameshttp://en.wikipedia.org/wiki/Moment_of_inertia#Body_frame_inertia_matrixhttp://en.wikipedia.org/wiki/Moment_of_inertia#Principal_axeshttp://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_ellipsoidhttp://en.wikipedia.org/wiki/Moment_of_inertia#See_alsohttp://en.wikipedia.org/wiki/Moment_of_inertia#Referenceshttp://en.wikipedia.org/wiki/Moment_of_inertia#External_linkshttp://en.wikipedia.org/wiki/Flywheelhttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/United_States_customary_unitshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-mach-1http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-mach-1http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Euler1730-2http://en.wikipedia.org/wiki/Euler's_laws#Euler.27s_second_lawhttp://en.wikipedia.org/wiki/Euler's_laws#Euler.27s_second_lawhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Symon_1971-4


3/20

$oment of inertia also appears inmomentum%kinetic energ"% and inNewton2s laws of motionfor a rigid bod"

as a ph"sical parameter that combines its shape and mass. There is an interesting difference in the wa" momentof inertia appears in planar and spatial movement. 5lanar movement has a single scalar that defines the moment

of inertia% while for spatial movement the same calculations "ield a )x) matrix of moments of inertia% called the

inertia matrix or inertia tensor.=0>=3>

The moment of inertia of a rotating fl"wheel is used in a machine to resist variations in applied tor#ue in orderto smooth its rotational output. The moment of inertia of an airplane about its longitudinal% hori,ontal and

vertical axes determines how steering forces on the control surfaces of its wings% elevators and tail affect theplane in roll% pitch and "aw.

Simple pendulum

A simple pendulum is a point mass suspended b" a string so that its movement is constrained to a circle arounda pivot point. The mass of a simple pendulum supported b" a light string accelerates due to the force of gravit".

The moment of inertia of the pendulum about the pivot point is its resistance to movement due to the tor#ue due

to gravit". $athematicall"% it is the ratio of the tor#ue due to gravit" about the pivot of a pendulum to its

angular acceleration about that pivot point. 'or a simple pendulum this is found to be the product the mass of

the particle with the s#uare of its distance to the pivot. This is shown as follows@

The force of gravit" on the mass of a simple pendulum generates a tor#ue around the axis

perpendicular to the plane of the pendulum movement.


4/20

Compound pendulum

5endulums used in $endenhall gravimeterapparatus% from 186 scientific &ournal. The portable gravimeter

developed in 18! b" Thomas . $endenhall provided the most accurate relative measurements of the localgravitational field of the +arth.

A compound pendulumis a bod" formed from an assembl" of particles or continuous shapes that rotates rigidl"

around a pivot. (ts moments of inertia is the sum the moments of inertia of each of the particles that is iscomposed of.=6>=8>@)0)3=>@010)The naturalfre#uenc" : of a compound pendulum depends on its moment of

inertia% %

where is the mass of the ob&ect% is local acceleration of gravit"% and is the distance from the pivot point tothe centre of mass of the ob&ect. $easuring this fre#uenc" of oscillation over small angular displacements

provides an effective wa" of measuring moment of inertia of a bod".=1!>@013016

Thus% to determine the moment of inertia of the bod"% simpl" suspend it from a convenient pivot point so that

it swings freel" in a plane perpendicular to the direction of the desired moment of inertia% then measure itsnatural fre#uenc" or period of oscillation :% to obtain

where is the period duration: of oscillation usuall" averaged over multiple periods:.

The moment of inertia of the bod" about its center of mass% % is then calculated using theparallel axistheoremto be

where is the mass of the bod" and is the distance from the pivot point to the center of mass .

$oment of inertia of a bod" is often defined in terms of itsradius of gyration%which is the radius of a ring ofe#ual mass around the center of mass of a bod" that has the same moment of inertia. The radius of g"ration is

calculated from the bod"2s moment of inertia and mass as the length%=11>@1-31-6
http://en.wikipedia.org/wiki/Gravimeterhttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Resnick-8http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Resnick-8http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-9http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-9http://en.wikipedia.org/wiki/Resonancehttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/File:Mendenhall_gravimeter_pendulums.jpghttp://en.wikipedia.org/wiki/File:Mendenhall_gravimeter_pendulums.jpghttp://en.wikipedia.org/wiki/Gravimeterhttp://en.wikipedia.org/wiki/Compound_pendulumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Resnick-8http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-9http://en.wikipedia.org/wiki/Resonancehttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Radius_of_gyrationhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11


5/20

Center of oscillation

A simple pendulum that has the same natural fre#uenc" as a compound pendulum defines the length from thepivot to a point called the center of oscillationof the compound pendulum. This point also corresponds to the

center of percussion. The length is determined from the formula%

or

The seconds pendulum% which provides the EtickE and EtockE of a grandfather clock% takes one second to swingfrom side*to*side. This is a period of two seconds% or a natural fre#uenc" of F radiansGsecond for the pendulum.

(n this case% the length is given b"%

Notice that the centre of oscillation of the seconds pendulum must be ad&usted to accommodate use in locations

with different values for the local acceleration of gravit". ater2s pendulumis an example of a compound

pendulum that is used to measure gravit" called a gravimeter.

Measuring moment of inertia

The moment of inertia of complex s"stems such as a vehicle or airplane around its vertical axis can be measured

b" suspending the s"stem from three points to form a trifilar pendulum. A trifilar pendulum is a platform

supported b" three wires designed to oscillate in torsion around its vertical centroidal axis.=1->The period ofoscillation of the trifilar pendulum "ields the moment of inertia of the s"stem.=1)>

Definition

The moment of inertia% (% is defined as the ratio of an applied tor#ue to the angular acceleration along a principalaxis of the ob&ect% where then H% and ( are scalars% that is

An e#uivalent definition of ( uses the angular momentumLas follows%

where is the angular velocit"of the ob&ect.
http://en.wikipedia.org/wiki/Center_of_oscillationhttp://en.wikipedia.org/wiki/Center_of_percussionhttp://en.wikipedia.org/wiki/Seconds_pendulumhttp://en.wikipedia.org/wiki/Kater's_pendulumhttp://en.wikipedia.org/wiki/Gravimeterhttp://en.wikipedia.org/wiki/Gravimeterhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-12http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-13http://en.wikipedia.org/wiki/Scalar_(physics)http://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Center_of_oscillationhttp://en.wikipedia.org/wiki/Center_of_percussionhttp://en.wikipedia.org/wiki/Seconds_pendulumhttp://en.wikipedia.org/wiki/Kater's_pendulumhttp://en.wikipedia.org/wiki/Gravimeterhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-12http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-13http://en.wikipedia.org/wiki/Scalar_(physics)http://en.wikipedia.org/wiki/Angular_velocity


6/20

Appl" this definition to a simple pendulum to see that the moment of inertia of the mass mabout the pivot point

at a distance ris

This generali,es to define the moment of inertia of a bod" about an axis Sas the sum of all elemental point

masses dmeach multiplied b" the s#uare of its perpendicular distance rto the axis.

Calculating moment of inertia about an axis

'our ob&ects racing down a plane while rolling without slipping. 'rom back to front@ spherical shell red:% solid

sphere orange:% c"lindrical ring green: and solid c"linder blue:. The time for each ob&ect to reach the

finishing line depends on their moment of inertia. Details% Animated I(' version:

The moment of inertia about an axis of a bod" is calculated b" summing mr-for ever" particle in the bod"%where ris the perpendicular distance to the specified axis. (n order to see how moment of inertia arises in the

stud" of the movement of an extended bod"% it is convenient to consider a rigid assembl" of point masses. This

e#uation can be used for axes that are not principal axes provided that it is understood that this does not full"describe the moment of inertia.=1/>:

onsider the kinetic energ" of an assembl" ofNmasses mithat lie at the distances rifrom the pivot pointP%

which is the nearest point on the axis of rotation. (t is the sum of the kinetic energ" of the individual masses%=1!>@013016=11>@1!8/1!80=11>@1-31)!!

This shows that the moment of inertia of the bod" is the sum of each of the mr-terms% that is

Thus% moment of inertia is a ph"sical propert" that combines the mass and distribution of the particles aroundthe rotation axis. Notice that rotation about different axes of the same bod" "ield different moments of inertia.

The moment of inertia of a continuous bod" rotating about a specified axis is calculated in the same wa"% with

the summation replaced b" the integral%
http://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gifhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-14http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-14http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Multiple_integralhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvhttp://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gifhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-14http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Multiple_integral


7/20


8/20

The moment of inertia of the compound pendulum is now obtained b" adding the moment of inertia of

the rod and the disc around the pivot pointPas%

whereLis 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 inertiaformulas for standard bod" shapes provides a wa" to obtain the moment of inertialof a complex bod" as an assembl" of simpler shaped bodies. Theparallel axis theoremis used to shift the

reference point of the individual bodies to the reference point of the assembl".

As one more example% consider the moment of inertia of a solid sphere of constant densit" about an axisthrough its center of mass. This is determined b" summing the moment of inertias of the thin discs that form the

sphere. (f the surface of the ball is defined b" the e#uation=11>@1)!1

then the radius rof the disc at the cross*section , along the ,*axis is

Therefore% the moment of inertia of the ball is the sum of the moment of inertias of the discs along the ,*axis%

where mB !"#$R#is the mass of the ball.

Moment of inertia in planar movement of a rigid body

(f a mechanical s"stemis constrained to move parallel to a fixed plane% then the rotation of a bod" in the s"stem

occurs around an axis %perpendicular to this plane. (n this case% the moment of inertia of the mass in this s"stemis a scalar known as thepolar moment of inertia. The definition of the polar moment of inertia can be obtained

b" considering momentum% kinetic energ" and Newton2s laws for the planar movement of a rigid s"stem of

particles.=6>=1!>=13>=16>

(f a s"stem of nparticles%Pi% iB 1%...%n% are assembled into a rigid bod"% then the momentum of the s"stem can

be written in terms of positions relative to a reference point R% and absolute velocities vi
http://en.wikipedia.org/wiki/List_of_moments_of_inertiahttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Mechanical_systemhttp://en.wikipedia.org/wiki/Mechanical_systemhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-17http://en.wikipedia.org/wiki/File:Moment_of_inertia_solid_sphere.svghttp://en.wikipedia.org/wiki/File:Moment_of_inertia_solid_sphere.svghttp://en.wikipedia.org/wiki/List_of_moments_of_inertiahttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Mechanical_systemhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-17


9/20

where K is the angular velocit" of the s"stem and is the velocit" of R.

'or planar movement the angular velocit" vector is directed along the unit vector %which is perpendicular to the

plane of movement. (ntroduce the unit vectors eifrom the reference point Rto a point ri% and the unit vector tiB

%C eiso

This defines the relative position vector and the velocit" vector for the rigid s"stem of the particles moving in a

plane.

Note on the cross product@ hen a bod" moves parallel to a ground plane% the tra&ectories of all the points in

the bod" lie in planes parallel to this ground plane. This means that an" rotation that the bod" undergoes mustbe around an axis perpendicular to this plane. 5lanar movement is often presented as pro&ected onto this ground

plane so that the axis of rotation appears as a point. (n this case% the angular velocit" and angular acceleration of

the bod" are scalars and the fact that the" are vectors along the rotation axis is ignored. This is usuall" preferredfor introductions to the topic. 7ut in the case of moment of inertia% the combination of mass and geometr"

benefits from the geometric properties of the cross product. 'or this reason% in this section on planar movementthe angular velocit" and accelerations of the bod" are vectors perpendicular to the ground plane% and the cross

product operations are the same as used for the stud" of spatial rigid bod" movement.

!ngular momentum in planar movement

A figure skater can reduce her moment of inertia b" pulling in her arms% allowing her to spin faster due toconservation of angular momentum.

The angular momentum vector for the planar movement of a rigid s"stem of particles is given b"=6>=1!>
http://en.wikipedia.org/wiki/Conservation_of_angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10http://en.wikipedia.org/wiki/File:Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpghttp://en.wikipedia.org/wiki/File:Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpghttp://en.wikipedia.org/wiki/Conservation_of_angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-B-Paul-7http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Uicker-10


10/20


11/20


12/20

This "ields the resultant tor#ue on the s"stem as

where eiC eiB !% and eiC tiB %is the unit vector perpendicular to the plane for all of the particlesPi.

;se the center of massCas the reference point and define the moment of inertia relative to the center of massI&% then the e#uation for the resultant tor#ue simplifies to

=11>@1!-

The parameterI&is thepolar moment of inertiaof the moving bod".

&he inertia matrix for spatial movement of a rigid body

The scalar moments of inertia appear as elements in a matrix when a s"stem of particles is assembled into a

rigid bod" that moves in three dimensional space. This inertia matrix appears in the calculation of the angularmomentum% kinetic energ" and resultant tor#ue of the rigid s"stem of particles. =)>=/>=0>=3>=18>

An important application of the inertia matrix and Newton2s laws of motion is the anal"sis of a spinning top.This is discussed in the article on I"roscopic precession.A more detailed presentation can be found in the

article on +uler2s e#uations of motion.

?et the s"stem of particlesPi% iB 1%...% nbe located at the coordinates riwith velocities virelative to a fixed

reference frame. 'or a possibl" moving: reference point R% the relative positions are

and the absolute: velocities are

where 'is the angular velocit" of the s"stem% and Ris the velocit" of R.

!ngular momentum

(f the reference pointRin the assembl"% or bod"% is chosen as the center of mass &% then its angular momentumtakes the form%=)>=3>

where the terms containing Rsum to ,ero b" definition of thecenter of mass.

(n order to define the inertia matrix% introduce the skew*s"mmetric matrix =)> constructed from a vector bthatperforms the cross product operation% such that
http://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Symon_1971-4http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Tenenbaum_2004-5http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Tsai-18http://en.wikipedia.org/wiki/Gyroscopic_precessionhttp://en.wikipedia.org/wiki/Gyroscopic_precessionhttp://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Beer-11http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Symon_1971-4http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Tenenbaum_2004-5http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Tsai-18http://en.wikipedia.org/wiki/Gyroscopic_precessionhttp://en.wikipedia.org/wiki/Euler's_equations_(rigid_body_dynamics)http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Center_of_mass


13/20

This matrix =)> has the components of bB *+, *y,*-$as its elements% in the form

Now construct the skew*s"mmetric matrix =Mri>B =ri.&> obtained from the relative position vector MriBri* C%

and use this skew*s"mmetric matrix to define%

where =I&> defined b"

is the inertia matrix of the rigid s"stem of particles measured relative to the center of mass &.

"inetic energy

The kinetic energ" of a rigid s"stem of particles can be formulated in terms of the center of massand a matrix

of mass moments of inertia of the s"stem. ?et the s"stem of particlesPi% iB 1%...%nbe located at the coordinates

riwith velocities vi% then the kinetic energ" is=)>=3>

where MriB ri*Cis the position vector of a particle relative to the center of mass.

This e#uation expands to "ield three terms

The second term in this e#uation is ,ero because Cis the center of mass. (ntroduce the skew*s"mmetric matrix

=Mri> so the kinetic energ" becomes
http://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6


14/20

Thus% the kinetic energ" of the rigid s"stem of particles is given b"

where =(> is the inertia matrix relative to the center of mass and $ is the total mass.

Resultant tor'ue

The inertia matrix appears in the application of Newton2s second law to a rigid assembl" of particles. Theresultant tor#ue on this s"stem is%=)>=3>

where aiis the acceleration of the particle 5 i. The kinematicsof a rigid bod" "ields the formula for the

acceleration of the particle 5iin terms of the position Rand acceleration !of the reference point% as well as theangular velocit" vector K and angular acceleration vector of the rigid s"stem as%

;se the center of mass Cas the reference point% and introduce the skew*s"mmetric matrix =Mri>B=ri*> to

represent the cross product ri* C:x% in order to obtain

This calculation uses the identit"

obtained from the Lacobi identit" for the triplecross product.

Thus% the resultant tor#ue on the rigid s"stem of particles is given b"

where =(> is the inertia matrix relative to the center of mass.

(arallel axis theorem

$ain article@ 5arallel axis theorem
http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Kinematicshttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Parallel_axis_theoremhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Marion_1995-3http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Kinematicshttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Parallel_axis_theorem


15/20

The inertia matrix of a bod" depends on the choice of the reference point. There is a useful relationship between

the inertia matrix relative to the center of mass Cand the inertia matrix relative to another point R. Thisrelationship is called the parallel axis theorem.=)>=3>

onsider the inertia matrix =(4> obtained for a rigid s"stem of particles measured relative to a reference point R%

given b"

?et Cbe the center of mass of the rigid s"stem% then

where dis the vector from the center of mass Cto the reference point R. ;se this e#uation to compute theinertia matrix%

+xpand this e#uation to obtain

The first term is the inertia matrix =(> relative to the center of mass. The second and third terms are ,ero b"

definition of the center of mass C. And the last term is the total mass of the s"stem multiplied b" the s#uare ofthe skew*s"mmetric matrix =d> constructed from d.

The result is the parallel axis theorem%

where dis the vector from the center of mass Cto the reference point R.

#ote on the minus sign@ 7" using the skew s"mmetric matrix of position vectors relative to the reference point%

the inertia matrix of each particle has the form .m=r>-% which is similar to the mr-that appears in planarmovement.


16/20

?et a rigid assembl" of rigid s"stem ofNparticles%Pi% iB 1%...%N% have coordinates ri. hoose Ras a reference

point and compute the moment of inertia around an axis ? defined b" the unit vector Sthrough the referencepoint R. The moment of inertia of the s"stem around this line ?BROtSis computed b" determining the

perpendicular vector from this axis to the particle 5igiven b"

where =(> is the identit" matrix and =SST> is the outer product matrix formed from the unit vector Salong theline ?.

(n order to relate this scalar moment of inertia to the inertia matrix of the bod"% introduce the skew*s"mmetric

matrix =S> such that =S>yBSx y% then we have the identit"

which relies on the fact that Sis a unit vector.

The magnitude s#uared of the perpendicular vector is

The simplification of this e#uation uses the identit"

where the dot and the cross products have been interchanged. +xpand the cross products to compute

where =Mri> is the skew s"mmetric matrix obtained from the vector MrBri*R.

Thus% the moment of inertia around the line ? through Rin the direction Sis obtained from the calculation

or

where =(4> is the moment of inertia matrix of the s"stem relative to the reference point R.

This shows that the inertia matrix can be used to calculate the moment of inertia of a bod" around an" specified

rotation axis in the bod".

&he inertia tensor


17/20

The inertia matrix is often described as the inertia tensor% which consists of the same moments of inertia and

products of inertia about the three coordinate axes.=3>=13>The inertia tensor is constructed from the ninecomponent tensors% the s"mbol is thetensor product:

where ei% iB1%-%) are the three orthogonal unit vectorsdefining the inertial frame in which the bod" moves.

;sing this basis the inertia tensor is given b"

This tensor is of degree two because the component tensors are each constructed from two basis vectors. (n this

form the inertia tensor is also called the inertia *inor.

'or a rigid s"stem of particlesP%% %B 1%...%Neach of mass mkwith position coordinates rkBxk% "k% ,k:% the inertiatensor is given b"

where Eis the identit" tensor

The inertia tensor for a continuous bod" is given b"

where rdefines the coordinates of a point in the bod" and Jr: is the mass densit" at that point. The integral istaken over the volume Vof the bod". The inertia tensor is s"mmetric because ( i&B (&i.

The inertia tensor can be used in the same wa" as the inertia matrix to compute the scalar moment of inertia

about an arbitrar" axis in the direction n%

where the dot product is taken with the corresponding elements in the component tensors. A product of inertiaterm such as (1-is obtained b" the computation

and can be interpreted as the moment of inertia around the x*axis when the ob&ect rotates around the "*axis.

The components of tensors of degree two can be assembled into a matrix. 'or the inertia tensor this matrix isgiven b"%
http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Tensor_producthttp://en.wikipedia.org/wiki/Tensor_producthttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Tensor_producthttp://en.wikipedia.org/wiki/Unit_vector


18/20

(t is common in rigid bod" mechanics to use notation that explicitl" identifies the x% "% and , axes% such as (xx

and (x"% for the components of the inertia tensor.

Identities for a s)e$*symmetric matrix

(n order to compute moment of inertia of a mass around an axis% the perpendicular vector from the mass to the

axis is needed. (f the axis ? is defined b" the unit vector Sthrough the reference point R% then the perpendicularvector from the line ? to the point ris given b"

where =(> is the identit" matrix and =SST> is the outer product matrix formed from the unit vector Salong the

line ?. 4ecall that skew*s"mmetric matrix =S> is constructed so that =S>yBSx y. The matrix =(*SST> in thise#uation subtracts the component of MrBr*Rthat is parallel to S.

The previous sections show that in computing the moment of inertia matrix this operator "ields a similar

operator using the components of the vector Mrthat is

(t is helpful to keep the following identities in mind (n order to compare the e#uations that define the inertia

tensor and the inertia matrix.

?et =4> be the skew s"mmetric matrix associated with the position vector RBx% "% ,:% then the product in the

inertia matrix becomes

This can be viewed as another wa" of computing the perpendicular distance from an axis to a point% because thematrix formed b" the outer product =RRT> "ields the identif"

where =(> is the )x) identit" matrix.

Also notice% that


19/20

where trdenotes the sum of the diagonal elements of the outer product matrix% known as its trace.

&he inertia matrix in different reference frames

The use of the inertia matrix in Newton2s second law assumes its components are computed relative to axes

parallel to the inertial frame and not relative to a bod"*fixed reference frame.=3>=13>This means that as the bod"

moves the components of the inertia matrix change with time. (n contrast% the components of the inertia matrix

measured in a bod"*fixed frame are constant.

+ody frame inertia matrix

?et the bod" frame inertia matrix relative to the center of mass be denoted =(7>% and define the orientation of

the bod" frame relative to the inertial frame b" the rotation matrix =A>% such that%

where vectors yin the bod" fixed coordinate frame have coordinates xin the inertial frame. Then% the inertia

matrix of the bod" measured in the inertial frame is given b"

Notice that =A> changes as the bod" moves% while =(7> remains constant.

(rincipal axes

$easured in the bod" frame the inertia matrix is a constant real s"mmetric matrix. A real s"mmetric matrix has

the eigendecompositioninto the product of a rotation matrix =P> and a diagonal matrix =Q>% given b"

where

The columns of the rotation matrix =P> define the directions of the principal axes of the bod"% and the constants(1% (-and ()are called the principal moments of inertia. This result was first shown b" L. L. S"lvester 180-:%

and is a form of S"lvester2s law of inertia.=1>=-!>

'or bodies with constant densit" an axis of rotational s"mmetr" is a principal axis.

Inertia ellipsoid
http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrixhttp://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrixhttp://en.wikipedia.org/wiki/James_Joseph_Sylvesterhttp://en.wikipedia.org/wiki/Sylvester's_law_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-syl852-19http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-syl852-19http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-norm-20http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Kane-6http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-Goldstein-16http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrixhttp://en.wikipedia.org/wiki/James_Joseph_Sylvesterhttp://en.wikipedia.org/wiki/Sylvester's_law_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-syl852-19http://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-norm-20


20/20

An ellipsoid with the semi*principal diameters labeled a% b% and c.

The moment of inertia matrix in bod"*frame coordinates is a #uadratic form that defines a surface in the bod"

called 5oinsot2s ellipsoid.=-1>?et =Q> be the inertia matrix relative to the center of mass aligned with the principal

axes% then the surface

or

defines an ellipsoidin the bod" frame. rite this e#uation in the form%

to see that the semi*principal diameters of this ellipsoid are given b"

?et a point xon this ellipsoid be defined in terms of its magnitude and direction% xBRxRn% where nis a unit

vector. Then the relationship presented above% between the inertia matrix and the scalar moment of inertia (naround an axis in the direction n% "ields

Thus% the magnitude of a point xin the direction non the inertia ellipsoid is
http://en.wikipedia.org/wiki/Poinsot's_ellipsoidhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-21http://en.wikipedia.org/wiki/Ellipsoidhttp://en.wikipedia.org/wiki/File:Triaxial_Ellipsoid.jpghttp://en.wikipedia.org/wiki/File:Triaxial_Ellipsoid.jpghttp://en.wikipedia.org/wiki/Poinsot's_ellipsoidhttp://en.wikipedia.org/wiki/Moment_of_inertia#cite_note-21http://en.wikipedia.org/wiki/Ellipsoid

moment of inertia basics understanding

Documents