A straight rod of length L has one of its end | vedclass

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Question

(C) The mass per unit length is given by $\lambda = Ax$.Consider an infinitesimal element of the rod of length $dx$ at a distance $x$ from the origin.

Vedclass · Accepted Answer

$2L/3$

Vedclass · Answer

(C) The mass per unit length is given by $\lambda = Ax$.Consider an infinitesimal element of the rod of length $dx$ at a distance $x$ from the origin.The mass of this element is $dm = \lambda dx = Ax dx$.The total mass $M$ of the rod is given by the integral of $dm$ from $0$ to $L$:$M = \int_{0}^{L} Ax dx = A [\frac{x^2}{2}]_{0}^{L} = \frac{AL^2}{2}$.The position of the center of mass $x_{cm}$ is given by the formula:$x_{cm} = \frac{1}{M} \int_{0}^{L} x dm$.Substituting the values:$x_{cm} = \frac{1}{(AL^2/2)} \int_{0}^{L} x (Ax dx) = \frac{2}{AL^2} \int_{0}^{L} Ax^2 dx$.$x_{cm} = \frac{2}{AL^2} \cdot A [\frac{x^3}{3}]_{0}^{L} = \frac{2}{L^2} \cdot \frac{L^3}{3} = \frac{2L}{3}$.Thus,the center of mass is at $2L/3$ from the origin.

$A$ straight rod of length $L$ has one of its ends at the origin and the other at $x = L$. If the mass per unit length of the rod is given by $\lambda = Ax$ (where $A$ is a constant),then where is its center of mass from the origin?

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