Question

The acceleration due to gravity is $g$ at a point distance $r$ from the center of earth $R$. if $r<R$ then

• A. $g\propto r$
• B. $g\propto r^2$
• C. $g\propto r^{-2}$
• D. $g\propto r^{-1}$

Answer 1

The correct option is A

$g\propto r$

Step 1: Given data

Acceleration due to gravity=$g$

Distance from the earth's center=$r$

Radius of earth=$R$

Relation between distance and radius= $r<R$

Step 2: Formula used:

Acceleration due to gravity at a depth $d$ below the surface of the earth is given using the formula-

$g\text{'}=g\left(1-\frac{d}{R}\right)$

Where

$g$ i acceleration due to gravity and $g\text{'}$ is the acceleration due to gravity at a depth $d$ below the surface of the earth

Step 3: Calculating the acceleration due to gravity at a point distance $r$ from the center of earth

Consider the following figure:

It is given that $r<R$ then-

in this case $R-d=r$

Now putting the value in formula used-

$$\begin{gathered} g\text{'}=g\left(1-\frac{d}{R}\right) \\ g\text{'}=g\left(\frac{R-d}{R}\right) \\ g\text{'}=g\frac{r}{R} \end{gathered}$$

According to this formula, gravity$\left(g\right)$ is directly proportional to the distance$\left(r\right)$. So, it can be written that, $g\propto r$.

Hence, option A is the correct answer.