A hypothetical spherical planet of radius R has density ρ(r) = Kr, where K is constant. The pressure at radius r

Question

A hypothetical spherical planet of radius R has density ρ(r) = Kr, where K is constant. The pressure at radius r < R is P(r) = [K²Gπ/ n 2 n^2 n 2 ](R⁴ - r⁴). Find n.

G1 Physics · Accepted Answer

Mass inside radius r is M(r)=4π∫₀ʳ Kr'·r'²dr' = πKr⁴. Hence g(r)=GM(r)/r² = πGKr². Hydrostatic equilibrium givₑ dP/dr = -ρg = -πGK²r³. With P(R)=0, P(r)=πGK²∫ᵣᴿ r'³dr' = [πGK²/4](R⁴-r⁴). Therefore n=4.

A hypothetical spherical planet of radius R has density ρ(r) = Kr, where K is constant. The pressure at radius r &lt; R is P(r) = [K²Gπ/ n 2 n^2 n 2 ](R⁴ - r⁴). Find n.

A hypothetical spherical planet of radius R has density ρ(r) = Kr, where K is constant. The pressure at radius r < R is P(r) = [K²Gπ/ n 2 n^2 n 2 ](R⁴ - r⁴). Find n.