First slide

Moment of inertia

Question

Figure shows a uniform solid block of mass M and edge lengths a, b and c. Its M.I. about an axis through one edge and perpendicular (as shown) to the large face of the block is

Moderate

A

$\frac{M}{3} (a^{2} + b^{2})$
B

$\frac{M}{4} (a^{2} + b^{2})$
C

$\frac{7 M}{12} (a^{2} + b^{2})$
D

$\frac{M}{12} (a^{2} + b^{2})$

Solution

$(MI)_{CG} = M (\frac{a^{2} + b^{2}}{12})$
According to the theorem of parallel axis,
$\begin{matrix} (MI)_{required axis} = (MI)_{CG} + M (O A)^{2} \\ = M (\frac{a^{2} + b^{2}}{12}) + M (\frac{a^{2} + b^{2}}{4}) \\ = M (\frac{a^{2} + b^{2}}{3}) \end{matrix}$

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