Question

The density of a certain planet varies with radial distance as: rho(r)= rho0(1-αr/R0) where R0= 25.12 x 106 m is the radius of the planet, rho0= 3800.0 kg/m3 is its central density, and α = 0.24. What is the total mass of this planet ?

Answer 1

`2.07*10^(26)kg`

Step-by-step explanation:

Mass of the planet can be found by integrating the density and the volume.

The volume of the planet would be:

dV = 4πr²dr

dM= ρ(r) 4πr²dr

Total mass:

`\int_(0)^(M) dm=\int_(0)^(R_o) \rho(r)4\pi r^2dr\\ \Rightarrow \int_(0)^(M) dm=\int_(0)^(R_o) \rho_o(1-(\alpha r)/(R_o))4\pi r^2dr\\ \Rightarrow \int_(0)^(M) dm=4\pi \rho_o \int_(0)^(R_o)(r^2-(\alpha r^3)/(R_o))dr\\ \Rightarrow M= 4\pi (\rho_o(r^3)/(3)-(\alpha r^4)/(4R_o))`

Insert the limits and then substitute the values:

`M=4\pi \rho_o ((R_o^3)/(3)-(\alpha R_o^3)/(4))\\ M =4\pi \rho_oR_o^3 ((1)/(3)-(\alpha)/(4))\\ M= 4* 3.14 * 3800.0* (25.12*10^6)^3((1)/(3)-(0.24)/(4))\\ M=2.07*10^(26)kg`