Chapter 5, Problem 59P

Determine the mass of the Earth from the known period and distance of the Moon.

To determine

The mass of the Earth from the known period and distance of the Moon.

Answer to Problem 59P

Solution:

With the known values of the period anddistance between Earth and Moon, the mass of the Earth is estimated as $5.98\times {10}^{24}\text{kg}$ .

Explanation of Solution

Given:

Period of Moon, $T=27.4\text{days}$

Formula Used:

According to Kepler’s third law of planetary motion:

  ${\left(\frac{2\pi }{T}\right)}^2=\frac{GM}{R^3}$

Where, M is the mass of the planet, R is the orbital radius, G is the gravitational constant and T is orbital period.

Calculation:

Convert units of period from days to seconds:

  $\begin{array}{l}\text{1}\text{day = 24}\text{h}\\ \text{1}\text{h = 3600 s}\end{array}$

  $\begin{array}{l}T=27.4\text{days}\\ T=27.4\times 24\times 3600\text{s}\end{array}$

The distance between Earth and Moon, $R=3.84\times {10}^8\text{m}$

According to Kepler’s law,

  ${\left(\frac{2\pi }{T}\right)}^2=\frac{GM}{R^3}$

  $M=\frac{4{\pi }^2{R}^3}{G{T}^2}=\frac{4{\pi }^2{\left(3.84\times {10}^8\right)}^3}{6.67\times {10}^{-11}\times {\left(27.4\times 24\times 3600\right)}^2}=5.98\times {10}^{24}\text{kg}$

Conclusion:

Therefore, with the known values of the periodand the distance between Earth and Moon, the mass of the Earth is estimated as $5.98\times {10}^{\text{24}}\text{kg}$ .