A thin rod of length L is placed along the X- | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The center of mass $x_{cm}$ is given by the formula:$x_{cm} = \frac{\int x dm}{\int dm} = \frac{\int_{0}^{L} x \cdot k(x/L)^n dx}{\int_{0}^{L} k(x

Vedclass · Accepted Answer

Vedclass · Answer

(D) The center of mass $x_{cm}$ is given by the formula:$x_{cm} = \frac{\int x dm}{\int dm} = \frac{\int_{0}^{L} x \cdot k(x/L)^n dx}{\int_{0}^{L} k(x/L)^n dx} = \frac{\int_{0}^{L} x^{n+1} dx}{\int_{0}^{L} x^n dx}$Evaluating the integrals:$x_{cm} = \frac{L^{n+2} / (n+2)}{L^{n+1} / (n+1)} = L \left( \frac{n+1}{n+2} ight)$Analyzing the behavior:$1$. When $n = 0$,$x_{cm} = L(1/2) = L/2$.$2$. As $n 	o \infty$,$x_{cm} = L \left( \frac{1 + 1/n}{1 + 2/n} ight) 	o L$.$3$. The derivative $\frac{dx_{cm}}{dn} = L \left( \frac{(n+2) - (n+1)}{(n+2)^2} ight) = \frac{L}{(n+2)^2} > 0$,which means $x_{cm}$ is an increasing function of $n$.Thus,the graph starts at $L/2$ for $n=0$ and asymptotically approaches $L$ as $n$ increases. This corresponds to the graph shown in option $D$.

Similar Questions

Consider a two-particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the center of mass through a distance $d$,by what distance should the second particle be moved so as to keep the center of mass at the same position?

Two objects of mass $10\,kg$ and $20\,kg$ respectively are connected to the two ends of a rigid rod of length $10\,m$ with negligible mass. The distance of the center of mass of the system from the $10\,kg$ mass is:

Four particles of masses $m_1 = 2m$,$m_2 = 4m$,$m_3 = m$,and $m_4$ are placed at the four corners of a square. What should be the value of $m_4$ so that the centre of mass of all the four particles is exactly at the centre of the square?

$A$ slender uniform rod of length $L$ is balanced vertically at a point $P$ on a horizontal surface having some friction. If the top of the rod is displaced slightly to the right,the position of its centre of mass at the time when the rod becomes horizontal is:

Is the statement "$A$ particle has point volume" true or false?

Vedclass Test Series

Exam Paper Generator

Online Exam Module