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A rod of length L is placed along the x-axis with one of its ends at the origin O. The mass of the rod increases linearly as $\frac{d m}{d x} = λ x$ where $λ$ is a constant

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The moment of inertia $y$ -axis is $\frac{1}{16} λ L^{4}$

The moment of inertia of rod about y-axis is $\frac{λ L^{4}}{4}$

The position of the centre of mass C of the rod, does not coincides with the geometrical centre C of the rod

All the above statements are wrong

answer is B, C.

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Detailed Solution

$\frac{d m}{d x} = λ x, d m = λ x d x$

$I_{y - a x i s} = \int d I = \int d m x^{2} = \int_{0}^{L} (λ x d x) x^{2} = λ {[\frac{x^{4}}{4}]}_{0}^{L} = \frac{λ L^{4}}{4}$

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