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there is a cubic frame,on each corner a pont mass'm' are kept,tell  about momet of inertia if axis is  pass through body diagonal of cube.each side of cube is 'a'.

Asked by Kunal 5th December 2017, 4:49 PM
Answered by Expert
Answer:
Let m1, m2,.....m8 are the point masses placed at each corner of the cube.  let r be the perpendicular distance from m1 to axis of rotation.
 
the distance r can be calculated from the triangle area shown in figure and the required equations are given below

begin mathsize 12px style capital delta space equals space square root of s left parenthesis s minus square root of 3 a right parenthesis left parenthesis s minus a right parenthesis left parenthesis s minus square root of 2 a right parenthesis end root space equals space 1 half open parentheses square root of 3 a close parentheses space r
w h e r e space s space equals fraction numerator a left parenthesis square root of 3 plus square root of 2 plus 1 right parenthesis over denominator 2 end fraction space
r space equals space fraction numerator 2 capital delta over denominator square root of 3 a end fraction space end style
 
as per my calculation I am getting r = (0.816)a
 
m2 and m8 are placed in the axis itself, hence do not contribute for moment of Inertia.
 
for all other 6 masses the perpendicular distance can be calculated as explained above and all the perpendicular distances are identical.
 
hence required Moment of Inertia = 6 × m × (0.816)2 × a2 = 3.995 ma2
Answered by Expert 6th December 2017, 3:29 PM
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