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(A) According to the perpendicular axis theorem,the moment of inertia of a planar body about an axis perpendicular to its plane and passing through it

Vedclass · Accepted Answer

$I_1 + I_2$

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(A) According to the perpendicular axis theorem,the moment of inertia of a planar body about an axis perpendicular to its plane and passing through its center is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane of the body and intersecting at the same point.In the given figure,$I_3$ and $I_4$ are two mutually perpendicular axes lying in the plane of the square plate and passing through its center $O$.Therefore,the moment of inertia about the axis perpendicular to the plane passing through $O$ is $I_z = I_3 + I_4$.Since the plate is a square,by symmetry,the moment of inertia about any two perpendicular axes in the plane passing through the center is equal. Thus,$I_3 = I_4$.Also,$I_1$ and $I_2$ are moments of inertia about the diagonal axes. For a square plate,the moment of inertia about any axis passing through the center in the plane is the same. Therefore,$I_1 = I_2 = I_3 = I_4$.Substituting these values,$I_z = I_3 + I_4 = I_1 + I_2$.

$A$ thin square plate $ABCD$ has a uniform thickness. What is the moment of inertia about an axis passing through its center and perpendicular to its plane?

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