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(A) For the elongation of a rod under its own weight, consider an element of length $dx$ at a distance $x$ from the lower end.The weight of the portio

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(A) For the elongation of a rod under its own weight, consider an element of length $dx$ at a distance $x$ from the lower end.The weight of the portion below this element is $W = (	ext{mass of portion}) 	imes g = (ho \cdot A \cdot x) \cdot g$, where $ho$ is the density, $A$ is the cross-sectional area, and $g$ is the acceleration due to gravity.The stress at this point is $\sigma = \frac{W}{A} = ho g x$.Using Hooke's Law, the strain is $\epsilon = \frac{\sigma}{Y} = \frac{ho g x}{Y}$, where $Y$ is Young's modulus.The elongation $dy$ for the element $dx$ is $dy = \epsilon dx = \frac{ho g x}{Y} dx$.To find the total elongation $y$ at a distance $x$ from the lower end, we integrate from $0$ to $x$:$y = \int_{0}^{x} \frac{ho g x'}{Y} dx' = \frac{ho g}{Y} \left[ \frac{x'^2}{2} ight]_{0}^{x} = \frac{ho g x^2}{2Y}$.Since $y \propto x^2$, the graph of elongation $(y)$ versus distance $(x)$ is an upward-opening parabola starting from the origin. Thus, the correct curve is the one shown in option $A$.

Which of the following curves represents the correct distribution of elongation $(y)$ along a heavy rod under its own weight? ($L$ = length of rod, $x$ = distance of a point from the lower end).

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