The mass density of a planet of radius R vari | vedclass

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

Question

(B) The gravitational field $E$ at a distance $r$ inside the planet is given by Gauss's Law for gravity: $E(4\pi r^{2}) = 4\pi G M(r)$,where $M(r)$ is

Vedclass · Accepted Answer

$r = \sqrt{\frac{5}{9}} R$

Vedclass · Answer

(B) The gravitational field $E$ at a distance $r$ inside the planet is given by Gauss's Law for gravity: $E(4\pi r^{2}) = 4\pi G M(r)$,where $M(r)$ is the mass enclosed within radius $r$.$M(r) = \int_{0}^{r} \rho(r) 4\pi r^{2} dr = 4\pi \rho_{0} \int_{0}^{r} \left(1 - \frac{r^{2}}{R^{2}}\right) r^{2} dr$.$M(r) = 4\pi \rho_{0} \left[ \frac{r^{3}}{3} - \frac{r^{5}}{5R^{2}} \right]$.Thus,$E = \frac{G M(r)}{r^{2}} = 4\pi G \rho_{0} \left( \frac{r}{3} - \frac{r^{3}}{5R^{2}} \right)$.To find the maximum field,set $\frac{dE}{dr} = 0$:$\frac{dE}{dr} = 4\pi G \rho_{0} \left( \frac{1}{3} - \frac{3r^{2}}{5R^{2}} \right) = 0$.$\frac{1}{3} = \frac{3r^{2}}{5R^{2}} \Rightarrow r^{2} = \frac{5R^{2}}{9} \Rightarrow r = \sqrt{\frac{5}{9}} R$.

The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $\rho(r) = \rho_{0} \left(1 - \frac{r^{2}}{R^{2}}\right)$. Then the gravitational field is maximum at:

Similar Questions

Two concentric shells of masses $m_1$ and $m_2$ have radii $r_2$ and $r_1$ respectively (where $r_2 < r_1$). $A$ particle of mass $m$ is placed at positions $A, B,$ and $C$ as shown in the figure. The gravitational forces on the particle at these positions are:

$A$ mass $m$ is placed at $P$ at a distance $h$ along the normal through the centre $O$ of a thin circular ring of mass $M$ and radius $r$. If the mass is moved further away such that $OP$ becomes $2h$,by what factor will the force of gravitation decrease,if $h = r$?

The gravitational force on a body of mass $1.5 \,kg$ situated at a point is $45 \,N$. The gravitational field intensity at that point is ......... $N/kg$.

If a cavity is made inside a solid sphere,then the gravitational field inside the cavity is

Two concentric hollow spheres of radius $R$ and $2R$ have the same mass $M$,as shown in the figure. The gravitational field intensity at point $P$ (where $R < r < 2R$) is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module