How Do You Calculate the Density Ratio of the Moon to the Earth?

[envscigrl]

problem:

The mass of the Earth is 5.98E+24 kg and its radius is 6370 km. The radius of the moon is 1738 km. The acceleration due to gravity at the surface of the moon is 1.62 m/s2. What is the ratio of the average density of the moon to that of the earth? (no units)
I am very confused as to how I find the density of planets given the equations and information I have. We are studying kepler's Laws and Newtons Law of Gravity. I feel like maybe I am missing some info or something.

Thanks for the help!

 

[Phymath]

density will be denoted p
[tex]p= \frac{m}{v}[/tex]
[tex]m_2a = \frac{G m_1 m_2}{r^2}[/tex]

[tex]1.62 = \frac{G m_1}{(1738*10^3 m)^2}[/tex]
[tex] m_{moon} = m_1, \ \frac{(1.62)(1738*10^3)^2}{G} = m_1 = 7.33 * 10^{22} kg [/tex]
[tex]p_{earth} = \frac{(5.98 *10^{24})}{(\frac{4}{3}pi (6370)^2)} = 3.52*10^{16} kg/km [/tex]
[tex]p_{moon} = 5.79*10^{13} kg/km [/tex]
[tex] p_{earth} : p_{moon} = 43.34 \ p_{moons} \ to \ every \ 1 \ p_{earth} [/tex]

 

[wais]

To find the density of a planet, you first need to know its mass and volume. The mass of the Earth is given as 5.98E+24 kg and the radius is 6370 km. Using the formula for volume of a sphere (V=4/3πr^3), we can calculate the volume of the Earth to be approximately 1.08E+21 km^3.

Similarly, the mass of the moon is not given but we can use the acceleration due to gravity at its surface (1.62 m/s^2) to calculate its mass using Newton's Law of Gravity (F=ma). The mass of the moon is approximately 7.35E+22 kg.

Using the same formula for volume, we can calculate the volume of the moon to be approximately 2.2E+10 km^3.

Now, to find the density of each planet, we can use the formula density=mass/volume. For the Earth, the density would be 5.98E+24 kg/1.08E+21 km^3, which equals to approximately 5.52 g/cm^3.

For the moon, the density would be 7.35E+22 kg/2.2E+10 km^3, which equals to approximately 3.34 g/cm^3.

To find the ratio of the average density of the moon to that of the Earth, we divide the density of the moon by the density of the Earth. This gives us a ratio of approximately 0.61, meaning that the moon has about 61% of the average density of the Earth.

I hope this helps clarify the process of finding the density of planets using the given information. Keep in mind that the equations and laws you mentioned are used to calculate different aspects of planetary motion, but they can also be used to calculate other properties of planets, such as density.