Q. A thin rod of length $'L'$ 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 centre of mass of the rod is plotted against $'n'$, which of the following graphs best approximates the dependence of $x_{cm}$ on $n$
A) [image:7_image2.emf]
B) [image:7_image3.emf]
C) [image:7_image4.emf]
D) [image:7_image5.emf]
Explanation
(d) $x_{cm} = \frac{\int_{}^{}xdm}{\int_{}^{}{dm}},\ x_{cm} = \frac{\int_{0}^{L}{x\ k\left( \frac{x}{L} \right)^{n}dx}}{\int_{0}^{L}{k\ \left( \frac{x}{L} \right)^{n}dx}} = \left( \frac{n + 1}{n + 2} \right)L$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.