A uniform rod of length L and mass M is place | vedclass

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(B) For a uniform rod of length $L$ and mass $M$ placed along the $x$-axis with one end at the origin,the linear mass density $\lambda$ is constant an

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$L/2$

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(B) For a uniform rod of length $L$ and mass $M$ placed along the $x$-axis with one end at the origin,the linear mass density $\lambda$ is constant and is given by $\lambda = M/L$.The position of the centre of mass $X_{cm}$ is defined as:$X_{cm} = \frac{1}{M} \int x \, dm$Since the rod is uniform,$dm = \lambda \, dx = (M/L) \, dx$.Substituting this into the integral:$X_{cm} = \frac{1}{M} \int_{0}^{L} x \left( \frac{M}{L} \right) dx$$X_{cm} = \frac{1}{L} \int_{0}^{L} x \, dx$$X_{cm} = \frac{1}{L} \left[ \frac{x^2}{2} \right]_{0}^{L}$$X_{cm} = \frac{1}{L} \left( \frac{L^2}{2} - 0 \right) = \frac{L}{2}$.Thus,the centre of mass is at $L/2$.

$A$ uniform rod of length $L$ and mass $M$ is placed along the $x$-axis with one end at the origin. Find the $x$-coordinate of the centre of mass of the rod.

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