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Moment of India is a tensor quantity but consider as scalar why?

Asked by krishnasingh56541220 | 30 Jul, 2023, 01:33: PM

Expert Answer

Moment of Inertia of system of particles about an axis depends on the orientation of axes relative to system of particles.

Moment of inertia is expressed as

begin mathsize 14px style I space equals space open vertical bar table row cell I subscript x x end subscript end cell cell I subscript x y end subscript end cell cell I subscript x z end subscript end cell row cell I subscript y x end subscript end cell cell I subscript y y end subscript end cell cell I subscript y z end subscript end cell row cell I subscript z x end subscript end cell cell I subscript z y end subscript end cell cell I subscript z z end subscript end cell end table close vertical bar end style

. . . . . . . . . . . . . . (1)

Moment of inertia I expressed above is Moment of inertia tensor .

Where individual elements are expressed as

begin mathsize 14px style I subscript x x end subscript space equals space sum for i of m subscript i left parenthesis space y subscript i superscript 2 space plus z subscript i superscript 2 space right parenthesis end style

;

begin mathsize 14px style I subscript y y end subscript space equals space sum for i of m subscript i left parenthesis space z subscript i superscript 2 space plus x subscript i superscript 2 space right parenthesis end style

;

begin mathsize 14px style I subscript z z end subscript space equals space sum for i of m subscript i left parenthesis space x subscript i superscript 2 space plus y subscript i superscript 2 space right parenthesis end style

;

begin mathsize 14px style I subscript x y end subscript space equals space I subscript y x end subscript space equals space minus space sum for i of m subscript i x subscript i y subscript i end style

;

begin mathsize 14px style I subscript y z end subscript space equals space I subscript z y end subscript space equals space minus space sum for i of m subscript i y subscript i z subscript i end style

;

begin mathsize 14px style I subscript z x end subscript space equals space I subscript x z end subscript space equals space minus space sum for i of m subscript i z subscript i x subscript i end style

;

Given a unit vector

begin mathsize 14px style u with hat on top end style

in the direction of rotation axis , then moment of inertia about its axis is given by

begin mathsize 14px style I subscript u space equals space u with hat on top subscript T space I space u with hat on top end style

begin mathsize 14px style sum for i j of u subscript i space I subscript i j end subscript space u subscript j space end style

Moment of inertia tensor is symmetric matrix and therfore it can be diagonalised by orthagonal transformation of cartesian axes.

It is essentially a reorientation of the orthagonal axes system. If the coordinate system is reoriented in this way ,

then moment of inertia tensor becomes diagonal, i.e., it has the form

begin mathsize 14px style I space equals space open vertical bar table row cell I subscript x x end subscript end cell 0 0 row 0 cell I subscript y y end subscript end cell 0 row 0 0 cell I subscript z z end subscript end cell end table close vertical bar end style

.................... (2)

The three diagonal elements are called principal components of moment of inertia and the corresponding axes that lead

to this diagonal form are the principle axes of the system of particles for which we want to get moment of inertia.

Each individual elements , i.e., I_xx , I_yy and I_zz are just numbers. hence they are scalar .

Where as the whole matrix I is tensor .

Answered by Thiyagarajan K | 31 Jul, 2023, 03:05: PM

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