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A square formed by 4 identical rods of length l and mass m .moment of inertia about x y and z axis when a side lie on x axis and other on y axis 

Asked by gungunmishra102 10th October 2019, 10:21 AM
Answered by Expert
figure shows the square formed by four identical rods of length L and Mass M placed in XY plane.
Moment of inertia Iy  about y axis  = IOA + IAB + IBC + IOC 
where IOA is moment of inertia of rod OA abount y-axis. Similarly for others.
since roatation axis passing through OA, and rod is thin, IOA = 0
IAB = IOC = (1/3)ML2
Each point of rod BC is at equal distance from rotation axis. Hence IBC = ML2
Hence Moment of inertia about y-axis = Iy = (1/3)ML2 + (1/3)ML2 + ML2 = (5/3)ML2
(M is mass of single rod)
Moment of inertial of x-axis is also similar to that of y-axis , hence Ix = (5/3)ML2
Now let us consider Moment of inertia of the square formed by four rods about z-axis
For rod OA and OC, moment of inertia is (1/3) ML2
Let us calculate the moment of inertia of rod CB.
let us consider small length dl at a distance l from one end C
Moment of inertia of  small mass dm of this small length is dm (L2 + l2) ( refer figure for distance)
If mass is distributed uniformly, mass per unit length is M/L. Hence mass of dl length dm = (M/L) dl
dI = dm (L2 + l2) = (M/L) (L2 + l2 ) dl
Moment of inertia of whole rod is given by,
begin mathsize 14px style I space equals space integral d I space equals space M over L integral subscript 0 superscript L left parenthesis L squared plus space l squared right parenthesis space d l space space equals space 4 over 3 M L squared end style
Moment of inertia of rod AB is also calculated in same manner and is equal to (4/3)ML2
Hence moment of inertia of the square formed by 4 rods about z-axis,
Iz= (1/3) ML2 + (1/3) ML2 +(4/3) ML2 +(4/3) ML2 = (10/3)  ML2
(M is mass of single rod)
Answered by Expert 10th October 2019, 3:19 PM
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