General
Easy
Physics-

For the given uniform square lamina A B C D, whose centre is O

Physics-General

  1. I subscript A C end subscript equals blank I subscript E F end subscript    
  2. square root of 2 blank end root I subscript A C end subscript equals blank I subscript E F end subscript    
  3. I subscript A D end subscript equals blank 4 I subscript E F end subscript    
  4. I subscript A C end subscript equals blank square root of 2 I subscript E F end subscript    

    Answer:The correct answer is: I subscript A D end subscript equals blank 4 I subscript E F end subscriptLet the each side of square lamina is d.
    So, I subscript E F end subscript equals I subscript G H end subscript (due to symmetry)
    And I subscript A C end subscript equals I subscript B D end subscript (due to symmetry)
    Now, according to theorem of perpendicular axis,

    I subscript A C end subscript plus I subscript B D end subscript equals I subscript 0 end subscript
    rightwards double arrow 2 I subscript A C end subscript equals I subscript 0 end subscript (i)
    and I subscript E F end subscript plus I subscript G H end subscript equals I subscript 0 end subscript
    rightwards double arrow 2 I subscript E F end subscript equals I subscript 0 end subscript (ii)
    From Eqs. (i) and (ii), we get
    I subscript A C end subscript equals I subscript E F end subscript
    therefore I subscript A D end subscript equals I subscript E F end subscript plus fraction numerator m d to the power of 2 end exponent over denominator 4 end fraction
    equals fraction numerator m d to the power of 2 end exponent over denominator 12 end fraction plus fraction numerator m d to the power of 2 end exponent over denominator 4 end fraction open parentheses a s I subscript E F end subscript equals fraction numerator m d to the power of 2 end exponent over denominator 12 end fraction close parentheses
    So, I subscript A D end subscript equals fraction numerator m d to the power of 2 end exponent over denominator 3 end fraction equals 4 I subscript E F end subscript

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    Moment of inertia of the system about the centre of plane is given by
    I equals open square brackets fraction numerator 2 over denominator 5 end fraction cross times 1 cross times open parentheses 0.1 close parentheses to the power of 2 end exponent plus 1 cross times open parentheses 1 close parentheses to the power of 2 end exponent close square brackets
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    plus open square brackets fraction numerator 2 over denominator 5 end fraction cross times 4 cross times open parentheses 0.1 close parentheses to the power of 2 end exponent plus 4 cross times open parentheses 1 close parentheses to the power of 2 end exponent close square brackets
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    fraction numerator L subscript T o t a l end subscript over denominator L subscript B end subscript end fraction equals fraction numerator left parenthesis I subscript A end subscript plus I subscript B end subscript right parenthesis omega over denominator I subscript B end subscript. omega end fraction (as omega will be same in both cases)
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    equals fraction numerator r subscript A end subscript over denominator r subscript B end subscript end fraction plus 1 (as m subscript A end subscript r subscript A end subscript equals m subscript B end subscript r subscript B end subscript right parenthesis
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    equals 6
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