A thin bar of length L has a mass per unit length λ, that increases… | JEE Main 2014 (11 Apr, Shift 1)

Q9JEE Main 2014 · 11 APR · SHIFT 1Difficult

Continuous Bodies: Standard Results & Variable Density

A thin bar of length L has a mass per unit length $\lambda$ , that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is $\lambda$ o, then the distance of the center of mass from the lighter end is:

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Answer: (d)

$\lambda = \lambda_o + kx$ ; $M = \lambda_o L + \frac{kL^2}{2} \Rightarrow kL^2 = 2(M - \lambda_o L)$
$\int_0^L x\lambda\,dx = \frac{\lambda_o L^2}{2} + \frac{kL^3}{3} = \frac{\lambda_o L^2}{2} + \frac{2(M - \lambda_o L)L}{3}$
$x_{cm} = \frac{1}{M}\left(\frac{2ML}{3} - \frac{\lambda_o L^2}{6}\right) = \frac{2L}{3} - \frac{\lambda_o L^2}{6M}$
So the answer is (d).

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