Chapter 5, Problem 67GP

To determine

The height above the surface of the Earth at which the gravitational acceleration becomes half.

Answer to Problem 67GP

The height above the surface of the earth at which the gravitational acceleration becomes half the gravitational acceleration on the surface of the earth is $1600\text{km}$ .

Explanation of Solution

Given:

The acceleration above the surface of the earth is, $g_h=\frac{g}{2}$ .

Formula Used:

The expression to calculate the height above the surface of the earth at which the gravitational acceleration becomes half the gravitational acceleration on the surface of the earth is,

  $g_h=g\left(1-\frac{2h}{R}\right)$

Here,

  $R$ is the radius of the earth and its value is $6400\text{km}$ .

  $g_h$ is acceleration due to gravity at height $h$ .

  $g$ is the gravitational acceleration.

Calculation:

Substitute all the values in the above expression.

  $\begin{array}{c}\frac{g}{2}=g\left(1-\frac{2h}{6400\text{km}}\right)\\ 1-\frac{h}{3200\text{km}}=\frac{1}{2}\\ \frac{h}{3200\text{km}}=\frac{1}{2}\\ h=1600\text{km}\end{array}$

Conclusion:

Thus, the height above the surface of the earth at which the gravitational acceleration becomes half the gravitational acceleration on the surface of the earth is $1600\text{km}$ .