Chapter 8, Problem 2P

(8.1) Calculate the densities of Venus and Jupiter (use the masses and radii given in the appendix). How do these numbers compare with the density of rock (about 3 grams per cm³) and water (1 gram per cm³)? (Note: Be sure to convert kilometers to centimeters and kilograms to grams if you are expressing your answer in grams per cm³.)

To determine

The densities of Venus and Jupiter and compare these densities with rock and water.

Answer to Problem 2P

The densities of Venus and Jupiter is $5.25{\text{g/cm}}^3$ and $1.24{\text{g/cm}}^3$. Using this Venus has large density than that of rock, Jupiter is denser than water.

Explanation of Solution

Write the expression for density of the object.

  ${\rho }_{\text{Venus}}=\frac{M_{\text{Venus}}}{V}$        (I)

Here, ${\rho }_{\text{Venus}}$ is density of the Venus planet, $M_{\text{Venus}}$ is the mass of the Venus planet, and $V$ is the volume of the planet.

The volume of the Venus planet is,

  $V=\frac{4}{3}\pi R_{\text{Venus}}^3$        (II)

Here, $R_{\text{Venus}}$ is the radius of the Venus planet.

Substitute equation (II) in equation (I).

  ${\rho }_{\text{Venus}}=\frac{M_{\text{Venus}}}{\frac{4}{3}\pi R_{\text{Venus}}^3}$        (III)

Similarly the density of the Jupiter planet is,

  ${\rho }_{\text{Jupiter}}=\frac{M_{\text{Jupiter}}}{\frac{4}{3}\pi R_{\text{Jupiter}}^3}$        (IV)

Here, ${\rho }_{\text{Jupiter}}$ is density of the Jupiter planet, $M_{\text{Jupiter}}$ is the mass of the Jupiter planet, and $R_{\text{Jupiter}}$ is the radius of the Jupiter planet.

Conclusion:

Substitute $4.87\times {10}^{24}\text{kg}$ for $M_{\text{Venus}}$ and $6,051\text{km}$ for $R_{\text{Venus}}$ in equation (III) to find ${\rho }_{\text{Venus}}$.

  $\begin{array}{c}{\rho }_{\text{Venus}}=\frac{\left(4.87\times {10}^{24}\text{kg}\right)\left(\frac{{10}^3\text{g}}{1\text{kg}}\right)}{\frac{4}{3}\pi {\left[\left(6,051\text{km}\right)\left(\frac{{10}^5\text{cm}}{1\text{km}}\right)\right]}^3}\\ =\frac{\left(4.87\times {10}^{27}\text{g}\right)}{\frac{4}{3}\left(3.14\right){\left(6,051\times {10}^5\text{cm}\right)}^3}\\ =5.25{\text{g/cm}}^3\end{array}$

Substitute $1.9\times {10}^{27}\text{kg}$ for $M_{\text{Jupiter}}$ and $71,492\text{km}$ for $R_{\text{Jupiter}}$ in equation (IV) to find ${\rho }_{\text{Jupiter}}$.

  $\begin{array}{c}{\rho }_{\text{Jupiter}}=\frac{\left(1.9\times {10}^{27}\text{kg}\right)\left(\frac{{10}^3\text{g}}{1\text{kg}}\right)}{\frac{4}{3}\pi {\left[\left(71,492\text{km}\right)\left(\frac{{10}^5\text{cm}}{1\text{km}}\right)\right]}^3}\\ =\frac{\left(1.9\times {10}^{30}\text{g}\right)}{\frac{4}{3}\left(3.14\right){\left(71,492\times {10}^5\text{cm}\right)}^3}\\ =1.24{\text{g/cm}}^3\end{array}$

Therefore, the densities of Venus and Jupiter is $5.25{\text{g/cm}}^3$ and $1.24{\text{g/cm}}^3$. Using this Venus has large density than that of rock, Jupiter is denser than water.

Since the density of rock is about $3{\text{g/cm}}^3$, so that Venus has larger density than that of rock suggested that it is due to the iron core of rock material.

Since the density of water is about $1{\text{g/cm}}^3$, so that Jupiter is slightly denser than water, suggesting that it is largely made of lighter materials such as hydrogen.