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  • A uniform thin rod AB of length L has linear mass density <img alt="\mu (x)=a+\frac{bx}{L}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cmu%20%28x%29%3Da&plus;%5Cfrac%7Bbx%7D%7BL

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

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\begin{aligned} &\mu(x)=a+\frac{b x}{L}\\ &M=\int_0^L \mu(x) d x=a L+\frac{b L}{2} \end{aligned}

\begin{gathered} x_{CM}=\frac{1}{M} \int_0^L x \mu(x) d x=\frac{\frac{a L^2}{2}+\frac{b L^2}{3}}{a L+\frac{b L}{2}}=\frac{L\left(\frac{a}{2}+\frac{b}{3}\right)}{a+\frac{b}{2}} \\ x_{CM}=\frac{7 L}{12} \\ \frac{\frac{a}{2}+\frac{b}{3}}{a+\frac{b}{2}}=\frac{7}{12} \Rightarrow b=2 a \end{gathered}

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Divya Sharma

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