A rod AB of l is such that its linear density (mass per unit length)μ varies as μ = ab − x where x is the distance of the section from end A (and b > l). The distance of the centre of mass from end A is b − zllnbb−l where z is ______

XCM=∫x dm∫dm=∫0lxab−xdx∫0lab−xdx=∫0lxb−xdx∫0l1b−xdx=∫0l−xb−xdx∫0l1b−xdx=∫0lb−x−bb−xdx∫0l1b−xdx=l−blnb−lnlnb−lb−−x+blnb−x−10lln(b−x0l=−l−lnbb−l−b=llnbb−l−b=-l-blnb-l-lnblnb-l-lnb=−l−blnb−lblnb−lb= −l−lnbb−1−b=llnbb−l−b

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A rod AB of $l$ is such that its linear density (mass per unit length) $μ$ varies as $μ = \frac{a}{b - x}$ where x is the distance of the section from end A (and b > $l$ ). The distance of the centre of mass from end A is $b - \frac{z l}{l n (\frac{b}{b - l})}$ where $z$ is ______

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Detailed Solution

$X_{C M} = \frac{\int x d m}{\int d m} = \frac{\int_{0}^{l} x \frac{a}{(b - x)} d x}{\int_{0}^{l} \frac{a}{(b - x)} d x} = \frac{\int_{0}^{l} \frac{x}{(b - x)} d x}{\int_{0}^{l} \frac{1}{(b - x)} d x}$

$= \frac{\int_{0}^{l} \frac{- x}{(b - x)} d x}{\int_{0}^{l} \frac{1}{(b - x)} d x} = \frac{\int_{0}^{l} \frac{(b - x) - b}{(b - x)} d x}{\int_{0}^{l} \frac{1}{(b - x)} d x}$