Chapter 7, Problem 50A

To determine

The mass of the Saturn.

Answer to Problem 50A

  $5.638\times {10}^{26}\text{kg}$

Explanation of Solution

Given:

The orbital radius of the Saturn’s moon, Mimas, is $1.87\times {10}^8\text{m}$ and its orbital period is 23.0 h.

Formula used:

Period of a Planet orbiting the Sun is given by,

  $T=2\pi \sqrt{\frac{r^3}{G{m}_S}}$

Where, $G$ is the gravitational constant, $r$ is the orbital radius, and $m_S$ is the mass of the Sun given as $1.99\times {10}^{30}\text{kg}$ .

Calculation:

The value of gravitational constant is,

  $G=6.67\times {10}^{-11}\text{N}{\text{.m}}^{\text{2}}{\text{/kg}}^{\text{2}}$

Then, considering the motion of moon around the Saturn as the motion of a planet around the Sun, the mass of the Saturn can be calculated as,

  $\begin{array}{c}T=2\pi \sqrt{\frac{r^3}{G{m}_{Saturn}}}\\ 23.0\text{h}=2\pi \sqrt{\frac{{\left(1.87\times {10}^8\right)}^3}{6.67\times {10}^{-11}\times m_{Saturn}}}\\ 23.0\times \text{3600}\text{s}=2\pi \sqrt{\frac{0.980\times {10}^{35}}{m_{Saturn}}}\\ {\left(13184.71\right)}^2\text{}=\frac{0.980\times {10}^{35}}{m_{Saturn}}\\ m_{Saturn}=\frac{0.980\times {10}^{35}}{1.738\times {10}^8\text{}}\\ =5.638\times {10}^{26}\text{kg}\end{array}$

Conclusion:

Thus, the mass of the Saturn is $5.638\times {10}^{26}\text{kg}$ .