The mass per unit length of a rod of length l | vedclass

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(C) The center of mass $X_{cm}$ of a continuous body is given by the formula: $X_{cm} = \frac{\int x dm}{\int dm}$.Given $\lambda = \frac{M_0 x}{l}$,t

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$\frac{2l}{3}$

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(C) The center of mass $X_{cm}$ of a continuous body is given by the formula: $X_{cm} = \frac{\int x dm}{\int dm}$.Given $\lambda = \frac{M_0 x}{l}$,the mass element $dm$ is $\lambda dx = \frac{M_0 x}{l} dx$.Substituting this into the formula:$X_{cm} = \frac{\int_{0}^{l} x (\frac{M_0 x}{l} dx)}{\int_{0}^{l} (\frac{M_0 x}{l} dx)}$$X_{cm} = \frac{\frac{M_0}{l} \int_{0}^{l} x^2 dx}{\frac{M_0}{l} \int_{0}^{l} x dx}$$X_{cm} = \frac{[x^3/3]_{0}^{l}}{[x^2/2]_{0}^{l}}$$X_{cm} = \frac{l^3/3}{l^2/2} = \frac{l^3}{3} 	imes \frac{2}{l^2} = \frac{2l}{3}$.

The mass per unit length of a rod of length $l$ is given by $\lambda = \frac{M_0 x}{l}$,where $M_0$ is a constant and $x$ is the distance from one end of the rod. The position of the center of mass of the rod is:

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