Question

Consider two planets with uniform mass distributions. The mass density and the radius of planet 1 are rho1 and R1, respectively, and those of planet 2 are rho2 and R2. What is the ratio of their masses?

Answer 1

`(R_1^3 * \rho _1)/(R_2^3 * \rho _2)`

Step-by-step explanation:

The mass density of an object with uniform mass distribution its defined as

`\rho = (mass)/(volume)`.

So, if we know the volume, and, we can obtain the mass of the object:

`mass = volume * \rho`.

Now, we can take the planets as spheres, of course, this is only an approximation, but good enough for us. The volume of a sphere of radius r its:

`Volume_(sphere) = (4)/(3) \pi r^3`

So, for our planets, the mass its given by:

`mass = volume * \rho\\mass = (4)/(3) \pi r ^3 \rho`,

so

`mass_(planet1) = (4)/(3) \pi R_1^3 \rho_1`

`mass_(planet2) = (4)/(3) \pi R_2^3 \rho_2`

Now, we can take the ratio:

`(mass_(planet1))/(mass_(planet2)) = ( (4)/(3) \pi R_1 ^3 \rho_1 )/( (4)/(3) \pi R_2 ^3 \rho_2 )`

Now, we can just cancel the
`(4)/(3) \pi [\tex] that appear in both sides of our fractions, and finally obtain: </p><p>[tex](R_1^3 * \rho _1)/(R_2^3 * \rho _2)`.