Chapter 7.1, Problem 7PP

To determine

The mean distance of an artificial satellite form Earth’s center.

Answer to Problem 7PP

  $4.3\times {10}^4\text{km}$

Explanation of Solution

Given:

Orbital period of Moon, $T_M=27.3\text{days}$

Orbital radius of Moon, $r_M=3.9\times {10}^5\text{km}$

Orbital period of artificial satellite, $T_{AS}=1\text{days}$

Formula Used:

From Kepler’s third law of planetary motion:

  $T^2\propto r^3$

Where, T is the orbital period and r is the orbital radius.

Calculations:

Let the orbital radius of artificial satellite be,

  $\begin{array}{c}\frac{T_M{}^2}{T_{AS}{}^2}=\frac{r_M{}^3}{r_{AS}{}^3}\\ r_{AS}=r_M{\left(\frac{T_{AS}}{T_M}\right)}^{2/3}\\ =\left(3.9\times {10}^5\text{km}\right){\left(\frac{1\text{day}}{27.3\text{days}}\right)}^{2/3}\\ =4.3\times {10}^4\text{km}\end{array}$

Conclusion:

The mean distance of an artificial satellite form Earth’s center is $4.3\times {10}^4\text{km}$ .