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

Four particles of masses $1\,kg, 2 \,kg, 3 \,kg$ and $4\, kg$ are placed at the four vertices $A, B, C$ and $D$ of a square of side $1\, m$ as shown in the figure. The coordinates of the centre of mass of the particles are:

What is the torque acting on the center of mass of an object due to the gravitational force?

Three identical spheres of radius $r$ are placed on a horizontal surface such that they touch each other. These spheres are arranged such that their centers form an equilateral triangle. The center of mass of this system will be at ........

Infinite rods of uniform mass density and lengths $L, L/2, L/4, \dots$ are placed one upon another up to infinity as shown in the figure. Find the $x-$ coordinate of the centre of mass.

Vedclass Test Series

Exam Paper Generator

Online Exam Module