Question

A symmetric lamina of mass $M$ consists of a square shape with a semicircular section over of the edge of the square as shown in figure. The side of the square is $2a$. The moment of inertia of the lamina about an axis through its centre of mass and perpendicular to the plane is $1.6M{a}^2$. The moment of inertia of the lamina about the tangent $AB$ in the plane of the lamina is ...............

Answer 1

$UsingperpendicularaxistheoremtheMIofthelaminaaboutaxesx{x}^{\prime }andy{y}^{\prime }is{I}_{x{x}^{\prime }}+I_{y{y}^{\prime }}=I=1.6M{a}^2.But{I}_{x{x}^{\prime }}and{I}_{y{y}^{\prime }}aresymmetrictoeachotherso{I}_{x{x}^{\prime }}=I_{y{y}^{\prime }}\Rightarrow I_{x{x}^{\prime }}=0.8M{a}^{2}x{x}^{\prime }istheaxisintheplaneofthelaminaandparalleltoAB.Distancebetweenx{x}^{\prime }andABis2aThus,byparallelaxistheorem{I}_{AB}=I_{x{x}^{\prime }}+M{\left(2a\right)}^2=4.8M{a}^2$