The distance of the centre of mass from end A | vedclass

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

Question

(C) The mass of a small element $dx$ at distance $x$ is given by $dm = \rho \cdot dx = \rho_{0} \left(1 - \frac{x^{2}}{L^{2}}\right) dx$.The position

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(C) The mass of a small element $dx$ at distance $x$ is given by $dm = \rho \cdot dx = \rho_{0} \left(1 - \frac{x^{2}}{L^{2}}\right) dx$.The position of the centre of mass $X_{cm}$ is given by the formula:$X_{cm} = \frac{\int x \, dm}{\int dm}$First,calculate the total mass $M = \int_{0}^{L} \rho_{0} \left(1 - \frac{x^{2}}{L^{2}}\right) dx = \rho_{0} \left[ x - \frac{x^{3}}{3L^{2}} \right]_{0}^{L} = \rho_{0} \left( L - \frac{L}{3} \right) = \frac{2}{3} \rho_{0} L$.Next,calculate the integral $\int x \, dm = \int_{0}^{L} x \cdot \rho_{0} \left(1 - \frac{x^{2}}{L^{2}}\right) dx = \rho_{0} \int_{0}^{L} \left( x - \frac{x^{3}}{L^{2}} \right) dx = \rho_{0} \left[ \frac{x^{2}}{2} - \frac{x^{4}}{4L^{2}} \right]_{0}^{L} = \rho_{0} \left( \frac{L^{2}}{2} - \frac{L^{2}}{4} \right) = \frac{1}{4} \rho_{0} L^{2}$.Now,calculate $X_{cm} = \frac{\frac{1}{4} \rho_{0} L^{2}}{\frac{2}{3} \rho_{0} L} = \frac{1}{4} \cdot \frac{3}{2} L = \frac{3L}{8}$.Comparing this with $\frac{3L}{\alpha}$,we get $\alpha = 8$.

Similar Questions

For determining the centre of mass of a body,why is a body considered as being composed of multiple small mass elements?

The integral $\int \vec{r} dm = 0$ for a homogeneous body is defined for which point?

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of the position vector of the centre of mass of this system will be equal to the magnitude of which of the following vectors?

For a body,the center of volume is defined as $\frac{\int \vec{r} \, dV}{\int dV}$ over the complete body,where $dV$ is a small volume element of the body and $\vec{r}$ is the position vector of that small volume from the origin. Which of the following statements is correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module