A thin rod of length 'L' is lying along the x -axis with its ends at… | JEE Main 2008 (AIEEE)

Q10JEE Main 2008 · AIEEEDifficult

Continuous Bodies: Standard Results & Variable Density

A thin rod of length $'L'$ is lying along the $x$ -axis with its ends at $x = 0$ and $x = L$ . Its linear density (mass/length) varies with $x$ as $k\left(\frac{x}{L}\right)^n,$ where $n$ can be zero or any positive number. If the position $X_{CM}$ of the center of mass of the rod is plotted against $'n',$ which of the following graphs best approximates the dependence of $X_{CM}$ on $n$ ?

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Answer: (a)

$X_{CM} = \frac{\int_0^L x^{n+1}dx}{\int_0^L x^n dx} = \frac{n + 1}{n + 2}L$
At $n = 0$ : $X_{CM} = \frac{L}{2}$ (uniform rod).
As $n$ increases, $X_{CM}$ rises steadily and approaches $L$ as $n \to \infty$ , rising fast at first then levelling off.
So the answer is (a).

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