The newly discovered spherical planet has a radius of R = 4 Times 10^7 m and a v

Question

The newly discovered spherical planet has a radius of R = 4 Times 10^7 m and a vanable density. Geological evidence indicates it is most dense at its center (core) and least dense at its surface. The functional relationship for it's density has been determined as follows: rho(r) = A - B squareroot r with A = 14,000 kg/m^3 & B = 2kg/m^1/2 where 0 LE r LE R Modeling the planet as a series of concentric spherical shells, determine the following physical quantifies.,. with including the computational (calculated) results: The mass M(R) of the planet:

Explanation / Answer

take a volume element dv at a distance r from the centre.

dv = 4r2 dr

mass = density x volume

differential mass be dm

dm = x dv

dm = (A + Br1/2 ) x 4r2 dr

dm = 4Ar2 dr + 4Br5/2 dr

integrating from 0 < r < R,

M = (4/3)A R3 + (4x2/7)BR7/2  

M = (4/3) (14000) (4 x 107 )3 + (8/7)(2)(4 x 107 )7/2  

M = 3.753156023 x 1027 + 2.906576681 x 1027  = 6.659732704 x 1027 kg

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