Question

Determine the distance from the Earth’s center to a point outside the Earth where the gravitational acceleration due to the Earth is$\frac{\mathbf{1}}{\mathbf{10}}$of its value at the Earth’s surface.

Short answer

The distance from the Earth’s centre is $2.01\times {10}^7\mathrm{m}$.

Step-by-step solution

Step 1. Given Data

The gravitational acceleration due to the Earth is $\frac{1}{10}$of its value at the Earth’s surface.

Step 2. Understanding the relation of acceleration due to gravity

In this problem, the general relation of the acceleration due to gravity of the Earth will be used.

Step 3. Estimating the distance from the Earth’s centre

The relation of gravitational accelerationis given by,

$\begin{array}{c}g=\frac{1}{10}{g}_{\mathrm{s}}\\ G\frac{m_{\mathrm{E}}}{r^2}=\frac{1}{10}G\frac{m_{\mathrm{E}}}{{r_{\mathrm{E}}}^2}\\ r^2=10{r_{\mathrm{E}}}^2\end{array}$

Here, $m_{\mathrm{E}}$is the mass of the Earth, r is the distance from the centre, G is the gravitational constant and $r_{\mathrm{E}}$is the radius of the Earth.

On plugging the values in the above relation.

$\begin{array}{c}{r}^2=10{\left(6.38\times {10}^6\mathrm{m}\right)}^2\\ r=2.01\times {10}^7\mathrm{m}\end{array}$

Thus, $r=2.01\times {10}^7\mathrm{m}$is the required distance.