JEE Class main Answered
Moment of India is a tensor quantity but consider as scalar why?
![question image](https://images.topperlearning.com/topper/new-ate/639478964c6194d98792IMG20230730133206.jpg)
Asked by krishnasingh56541220 | 30 Jul, 2023, 13:33: PM
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](https://images.topperlearning.com/topper/tinymce/cache/afe175069aa555eb8a7c71405de17c9b.png)
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](https://images.topperlearning.com/topper/tinymce/cache/4650ac54119883ea4c7eaf3aaf7a57ef.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/842a4239c8dfad03014a7ffcddf2a6fb.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/e2f4d2ee8210a0d597f31a29863e5722.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/525334959b38ff13f42ceec88998ab1e.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/01d292e0290a65f70e94d2d3640f4141.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/154ea516ac058cae11c78605b738dc71.png)
Given a unit vector
in the direction of rotation axis , then moment of inertia about its axis is given by
![begin mathsize 14px style u with hat on top end style](https://images.topperlearning.com/topper/tinymce/cache/7ebd31d4eb7f4f5ac0a25c2dda407477.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/809be2ff9f42cbe226a0d4f123aa3618.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/b3bfa9d8878716db2ff61b5f276a528d.png)
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](https://images.topperlearning.com/topper/tinymce/cache/2dfd0815308615916f70ec7770a9f4a5.png)
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., Ixx , Iyy and Izz are just numbers. hence they are scalar .
Where as the whole matrix I is tensor .
Answered by Thiyagarajan K | 31 Jul, 2023, 15:05: PM
JEE main - Physics
Asked by aksharashere3 | 26 Jan, 2024, 19:04: PM
JEE main - Physics
Asked by patidargiriraj9 | 31 Dec, 2023, 21:12: PM
JEE main - Physics
Asked by adullanaipunya | 08 Sep, 2023, 19:41: PM
JEE main - Physics
Asked by krishnasingh56541220 | 30 Jul, 2023, 13:33: PM
JEE main - Physics
Asked by harshveer262006 | 09 Jan, 2023, 21:41: PM
JEE main - Physics
Asked by shanusunny15 | 26 Oct, 2022, 20:46: PM
JEE main - Physics
Asked by saragilgan | 06 Jun, 2022, 16:31: PM
JEE main - Physics
Asked by mayursamahajan | 09 Jun, 2021, 21:13: PM