A hypothetical spherical planet of radius R has density ρ(r) = Kr. The pressure at radius r

Question

A hypothetical spherical planet of radius R has density ρ(r) = Kr. The pressure at radius r < R is P(r) = [K²Gπ/n](R⁴ - r⁴). Find n.

G1 Physics · Accepted Answer

M(r)=4π∫₀ʳ Kr'·r'²dr'=πKr⁴, so g(r)=GM(r)/r²=πGKr². Hydrostatic equilibrium givₑ dP/dr=-ρg=-πGK²r³. With P(R)=0, P(r)=πGK²(R⁴-r⁴)/4. Hence n=4.

A hypothetical spherical planet of radius R has density ρ(r) = Kr. The pressure at radius r &lt; R is P(r) = [K²Gπ/n](R⁴ - r⁴). Find n.