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moment of Inertia

Asked by harshveer262006 9th January 2023, 9:41 PM
Answered by Expert
Fig.(1) shows the moment of inertia of disk of radius R with respect to the rotation axis perpendicular to the
plane of disk and passing through centre of mass.
Let the axis of rotation in fig.(1) coincide with z-axis of cartesian coordinate system.
Then begin mathsize 14px style I subscript z space equals space 1 half M space R squared end style
where M is mass of disc .
If our axis of rotation is either x-axis or y-axis as shown in fig.(2) ,
by perpendicular axis theorem
begin mathsize 14px style I subscript z equals space I subscript x space plus space I subscript y end style
By symmetry Ix = Iy
Hence Iy = (1/2) Iz = (1/2) (1/2) M R2 = (1/4) M R2
If axis of rotation is shifted to the edge of disc as shown in figure ,
Then by parallel axis theorem ,
begin mathsize 14px style I space equals space I subscript y space plus space M space R squared space equals space M R to the power of 2 to the power of space end exponent left parenthesis space 1 space plus space 1 fourth space right parenthesis space space equals space 5 over 4 M space R squared end style
Answered by Expert 10th January 2023, 12:03 AM
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