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What is the moment of inertia of the thi...

What is the moment of inertia of the thin uniform rectangular plate of mass m about the diagonal axis `xx'` ?

A

`(m(a^(2) + b^(2))/24)`

B

`(m.a^(2)b^(2))/(6(a^(2) + b^(2))`

C

`(m.a^(2)b^(2))/(12(a^(2) + b^(2))`

D

`(m(a^(2) + b^(2))/12)`

Text Solution

Verified by Experts

The correct Answer is:
B
M.I. of rectangle =2(M. I. of triangle).
Consider a rod element of length l and width dx `1/x =b/a rArr 1=b/a x`
mass of element `dm =m/(ab) 1.dx = m/(ab) (b/a x) dx = m/a^(2) x dx`
M.I. of the rod
`dI = ((dm)I^(2))/3 sin^(2)theta = (m/a^(2) xx dx)(b^(2)x^(2))/a^(2).(sin^(2)theta)/3 = (mb^(2))/(3a^(4)) sin^(4) theta x^(3)dx`

`I=int (dI) = (mb^(2))/(3a^(4)) sin^(2)theta int_(0)^(a) x^(3)dx = (mb^(2))/12 .a^(2)/(a^(2) + b^(2)),` M.I of rectangular plate = `2I=(ma^(2)b^(2))/(6(a^(2) + b^(2))`
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