Question

Assuming Earth, an uniform sphere of mass M and radius R. At a point P inside Earth at a distance r measured from centre of Earth ( r < R). Pressure (p) caused by gravitational compression is

• A. $\frac{3GM{r}^2}{8\pi R^6}$
• B. $\frac{3}{8}[1-\frac{r^2}{R^2}]\frac{Gm}{\pi R^3}$
• C. $\frac{3}{2}\frac{GM}{R^2}$
• D. Zero

Answer 1

The correct option is A $\frac{3GM{r}^2}{8\pi R^6}$

$\begin{array}{c}\frac{GMx}{R^3}\cdot dm\\ d_F=\frac{GMx}{R^3}\cdot \delta \cdot 4\pi x^{2}dx\\ d_F=\frac{GM\delta 4\pi }{R^3}{x}^{3}dx\\ d_p=\frac{GM\delta 4\pi }{R^3}\cdot \frac{x^{3}dx}{4\pi x^2}\\ =\frac{GM\delta }{R^3}\cdot \frac{r^2}{2}\\ =\frac{GM{r}^2}{2R^3}\cdot \frac{M}{\frac{4}{3}\pi R^3}\\ =\frac{3GM{r}^2}{8\pi R^6}\end{array}$

Hence,

option $\left(A\right)$ is correct answer.