Question

The radius of earth is $6400km$. At what depth below the earth surface does the acceleration due to gravity decreases by $1\%$?

• A. $64km$
• B. $44km$
• C. $10km$
• D. $150km$

Answer 1

The correct option is A

$64km$

Step 1: Calculating the acceleration due to gravity if decreased by $1\%$

The acceleration due to gravity on Earth's surface is $g$.

Let the decreased acceleration be $g\text{'}$.

According to the question,

$$\begin{gathered} g\text{'}=g-1\%ofg \\ g\text{'}=g-\frac{1}{100}g \\ g\text{'}=\frac{100g-1g}{100} \\ g\text{'}=\frac{99}{100}g \end{gathered}$$

Step 2: Calculating the depth below the earth's surface where the acceleration due to gravity decreases by $1\%$

The acceleration is due to gravity at depth $h$below the earth's surface $g\text{'}=g\left(1-\frac{h}{r}\right)$

where $r$ is the radius of earth.

Substituting the value of $g\text{'}$ from Step 1, we get,

$$\begin{gathered} \frac{99}{100}g=g\left(1-\frac{h}{r}\right) \\ \frac{99}{100}=1-\frac{h}{r} \\ \frac{h}{r}=1-\frac{99}{100} \\ \frac{h}{r}=\frac{100-99}{100} \\ \frac{h}{r}=\frac{1}{100} \end{gathered}$$

The value of $r$ is given $6400km$ in the question.

$$\begin{gathered} h=\frac{6400}{100}km \\ h=64km \end{gathered}$$

So, at a depth of $64km$below the earth's surface, the acceleration due to gravity decreases by $1\%$.

Hence, option A is correct.