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 variable density. Geological evidence indicates it is most dense at its center (core) and least dense at its surface. e functional relationship for it's density has been determined as follows: p(r) = A - B Squareroot r with A = 14,000 kg/m^3 & B = 2kg/m^7/2 where 0 lessthanorequalto r lessthanorequalto R Modeling the planet as a series of concentric spherical shells, determine the following physical quantities... with including the computational (calculated) results: The mass M(R) of the planet: The surface gravity g(R): The radial location r* (critical point) where the acceleration of its gravity is a maximum: The maximum value of its gravitational acceleration... g_max:

Explanation / Answer

radius R=4*10^7 m

density, rho(r)=A-B*sqrt(r)

rho=14000-2*sqrt(r)

a)

mass , M=rho*V

=(14000-2*sqrt(r))*(4/3*pi*r^3)

=(14000-2*sqrt(r))*(4/3*pi*r^3)

=(14000-2*sqrt(4*10^7))*(4/3*pi*(4*10^7)^3)

=3.62*10^26 kg

b)

g=G*m/r^2

=(6.67*10^-11*3.62*10^26)/(4*10^7)^2

=15.1 m/sec^2

c)

acceleeration due to gravity is maximu at the pole of the planet

d)

maximum value of g_max=15.1 m/sec^2

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