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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 $\;\lambda_{0}\;$ , then the distance of the centre of mass from the lighter end is :

$(a)\;\large\frac{L}{2}-\large\frac{\lambda_{0}L^{2}}{4M} \qquad(b)\;\large\frac{L}{3}+\large\frac{\lambda_{0}L^{2}}{8M}\qquad(c)\;\large\frac{L}{3}+\large\frac{\lambda_{0}L^{2}}{4M}\qquad(d)\;\large\frac{2L}{3}-\large\frac{\lambda_{0}L^{2}}{6M}$

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