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The moment of inertia of an uniform thin sheet of mass M of the given shape about the specified axis is (axis and sheet both are in same plane)

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Asked by m.nilu 19th October 2018, 12:30 PM
Answered by Expert
Answer:
Let us first calculate the moment of inertia of half square plate that is cut through the diagonal
and the axis of rotation is along the diagonal as shown in figure.1.

Let the axis of rotation is along Y-axis of cartesian coordinate system.
Let us consider a strip of mass dm whose length is 2y and its width is dx.
 
Moment of Inertia dI of this strip is given by, dI = dm × x2 ...............(1)

mass of strip dm = Area×density = 2y × dx × ρ = 2×[ ( a/√2 ) - x ] × dx × ρ .........................(2)
 
[ ΔAOB and ΔDEB are similar triangles. hence  DE/EB = AO/OB =1, hence DE = EB = ( a/√2 ) - x ) ]
where ρ is density per unit surface area.
 
Hence eqn.(1) becomes, dI = 2×[ ( a/√2 ) - x ] × x2 × dx × ρ
 
Moment of Inertial of half square plate I = begin mathsize 12px style integral d I space equals 2 rho space integral subscript 0 superscript fraction numerator a over denominator square root of 2 end fraction end superscript open parentheses fraction numerator a over denominator square root of 2 end fraction minus x close parentheses x squared d x space equals space fraction numerator rho a to the power of 4 over denominator 24 end fraction space.......................... left parenthesis 3 right parenthesis end style
Hence Moment of inertia of full square plate about the axis of rotation along its diagonal as shown in fig.(2) is given by,
2I =begin mathsize 12px style fraction numerator rho a to the power of 4 over denominator 12 end fraction space equals space rho a squared cross times a squared over 12 space equals space M cross times a squared over 12 space.................. left parenthesis 4 right parenthesis end style
Where M is mass of the given square plate.

If the axis of rotation is shifted to corner of the square as shown in fig.(3), then the required moment of inertia is given by,
begin mathsize 12px style 2 I space plus space M cross times a squared over 2 space equals space 1 over 12 M cross times a squared space plus M cross times a squared over 2 space equals space 7 over 12 space M cross times a squared end style
Answered by Expert 20th October 2018, 3:05 AM
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