Question

The acceleration due to gravity near the surface of a planet of radius $R$ and density $d$ is proportional to

• A. $\frac{d}{R^2}$
• B. $d{R}^2$
• C. $dR$
• D. $\frac{d}{R}$

Answer 1

The correct option is C

$dR$

Step 1: Given data:

Radius of planet = $R$

Density of planet = $d$

Step 2: Formula used

Acceleration due to gravity is $g=\frac{GM}{R^2}----\left(1\right)$

Where $G$ is the gravitational constant.

$M$ is the mass of the planet and R its radius.

$$\begin{gathered} Density=\frac{mass}{volume} \\ d=\frac{M}{V} \end{gathered}$$

Step 3: Finding the relationship

From the formula of volume considering the planet as a solid sphere,

$V=\frac{4}{3}\pi R^3$

We know that,

$M=d\times \frac{4}{3}{\mathrm{\pi R}}^3----\left(2\right)$

Substitute the value of mass from equation$\left(2\right)$ to equation (1) we get-

$g=\frac{G\times d\times {\displaystyle \frac{4}{3}}{\mathrm{\pi R}}^3}{R^2}$

$g=\frac{4}{3}\pi d\mathrm{GR}$

So, it can be concluded that, $g\propto dR$

Hence, option C is the correct answer.