The density of a solid sphere of radius R is | vedclass

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

Question

(D) The volume of a spherical shell of thickness $dr$ and radius $r$ is $dV = 4\pi r^2 dr$.The mass of this shell is $dM = \rho dV = \left( 20 \frac{r

Vedclass · Accepted Answer

$\pi$

Vedclass · Answer

(D) The volume of a spherical shell of thickness $dr$ and radius $r$ is $dV = 4\pi r^2 dr$.The mass of this shell is $dM = \rho dV = \left( 20 \frac{r^2}{R^2} \right) \cdot 4\pi r^2 dr = \frac{80\pi r^4}{R^2} dr$.The total mass $M$ of the solid sphere is obtained by integrating from $0$ to $R$:$M = \int_0^R dM = \int_0^R \frac{80\pi r^4}{R^2} dr = \frac{80\pi}{R^2} \left[ \frac{r^5}{5} \right]_0^R = \frac{80\pi R^5}{5R^2} = 16\pi R^3$.For a point outside the sphere at a distance $r = 4R$,the sphere acts as a point mass $M$ at its centre. The gravitational field $E$ is given by:$E = \frac{GM}{r^2} = \frac{G(16\pi R^3)}{(4R)^2} = \frac{16\pi G R^3}{16R^2} = \pi GR$.Therefore,the ratio $\frac{E}{GR} = \pi$.

The density of a solid sphere of radius $R$ is $\rho(r) = 20 \frac{r^2}{R^2}$,where $r$ is the distance from its centre. If the gravitational field due to this sphere at a distance $4R$ from its centre is $E$ and $G$ is the gravitational constant,then the ratio of $\frac{E}{GR}$ is

Similar Questions

$3$ particles each of mass $m$ are kept at the vertices of an equilateral triangle of side $L$. The gravitational field at the centre due to these particles is

There are two bodies of masses $100 \, kg$ and $10000 \, kg$ separated by a distance of $1 \, m$. At what distance from the smaller body will the intensity of the gravitational field be zero?

$A$ spherical shell is divided into two parts. If the gravitational intensity at point $P$ due to the upper part is ${I_1}$ and due to the lower part is ${I_2}$,then:

Infinite particles,each of mass $3 \ kg$,are placed at distances of $1 \ m, 2 \ m, 4 \ m, 8 \ m, \dots$ from point $O$. What is the gravitational intensity at point $O$?

$A$ spherical shell is cut into two pieces along a plane as shown in the figure. $P$ is a point on the plane of the cut. The gravitational field at $P$ due to the upper part is $I_1$ and that due to the lower part is $I_2$. What is the relation between them?

Vedclass Test Series

Exam Paper Generator

Online Exam Module