A uniform thin rod AB of length L has linear mass density μ(x)=a+bxL, where x is measured from A. If the CM of the rod lies at a distance of (712) L from A, then a and b are related as :

xcm=∫0L(dm)x∫0Ldm=∫0L(a+bxL)dx.x∫aL(a+bxL)dx

xcm=∫0L(dm)x∫0Ldm=∫0L(a+bxL)dx.x∫aL(a+bxL)dx

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A uniform thin rod AB of length L has linear mass density $μ (x) = a + \frac{b x}{L}$ , where $x$ is measured from A. If the CM of the rod lies at a distance of $(\frac{7}{12})$ L from A, then a and b are related as :

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$3 a = 2 b$

$a = 2 b$

$2 a = b$

$a = b$

answer is B.

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$x_{c m} = \frac{\int_{0}^{L} (d m) x}{\int_{0}^{L} d m} = \frac{\int_{0}^{L} (a + \frac{b x}{L}) d x . x}{\int_{a}^{L} (a + \frac{b x}{L}) d x}$

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