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A rod of length $L$ is of non uniform cross-section. Its mass per unit length varies linearly with distance from left end. Then its centre of mass from left end is at a distance. (Take linear density at left end $= 0$ )

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$\frac{2 L}{3}$

$\frac{L}{3}$

$\frac{L}{2}$

$\frac{L}{6}$

answer is D.

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Detailed Solution

Consider a small conical part at a distance x from left end.

$X_{c m} = \frac{\int x d m}{\int d m} X_{c m} = \frac{\int_{0}^{L} (λ_{x} d x) x}{\int_{0}^{L} λ_{x} d x} = \frac{2 L}{3} (∵ λ = \frac{d m}{d x} \Rightarrow λ_{x} = \frac{x λ}{L})$

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