Question

The gravitational field at a distance $R/2$ from the center of the earth is $E_1$. The gravitational field at a height $R/2$ from the surface of earth is $E_2$. Then $E_1/E_2$ is:

• A. $3:2$
• B. $2:3$
• C. $9:8$
• D. $3:8$

Answer 1

The correct option is D $3:8$

$E_1=\frac{{GM}^1}{{\left(\frac{R}{2}\right)}^2}=\frac{G\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3}{{\left(\frac{R}{2}\right)}^2}$

$E_2=\frac{GM}{{(R_1+R_2)}^2}=\frac{G\left(\frac{4}{3}\pi R^3\right)}{\frac{9R^2}{4}}$

$E_1=\frac{2G{R}_P\pi }{3}$

$E_2=\frac{16}{9}G{R}_P\pi$

$\Rightarrow \frac{E_1}{E_2}=\frac{2}{5}\times \frac{9}{16}=\frac{3}{8}$