Question

A spherical planet of radius R has spherically symmetrical distribution of mass density, varying as square of the distance from the centre, from zero at centre to maximum value ρ 0 at its surface.

Answer 1

The question is incomplete. Properties derived are: ρ(r) = (ρ₀/R²) * r², M = (4πρ₀R³)/5, g(r) = (GMr³)/R⁵ (r ≤ R), g(r) = GM/r² (r > R), V(r) = (5GM)/(4R) - (GMr⁴)/(4R⁵) (r ≤ R), V(r) = GM/r (r > R)

Answer 2

The question provides a description of a spherical planet with a specific mass density distribution but does not ask a specific question. Based on the context of typical physics problems involving mass distributions, common tasks are to find the density function, total mass, gravitational field, or gravitational potential. Let's derive these properties.

1. Density function ρ(r):
   The density varies as the square of the distance from the center, r. Let ρ(r) = C * r², where C is a constant.
   At the surface (r=R), the density is ρ₀.
   So, ρ(R) = C * R² = ρ₀.
   This gives C = ρ₀/R².
   The density function is ρ(r) = (ρ₀/R²) * r².

2. Total mass M of the planet:
   The total mass is the integral of the density over the volume of the sphere. Consider a spherical shell of radius r and thickness dr. The volume element is dV = 4πr²dr.
   The mass element is dm = ρ(r) * dV = ((ρ₀/R²) * r²) * (4πr²dr) = (4πρ₀/R²) * r⁴dr.
   The total mass is M = ∫₀ᴿ dm = ∫₀ᴿ (4πρ₀/R²) * r⁴dr = (4πρ₀/R²) * [r⁵/5]₀ᴿ = (4πρ₀/R²) * (R⁵/5) = (4πρ₀R³)/5.

3. Gravitational field g(r) at a distance r from the center:

   • Inside the planet (r ≤ R):
     The gravitational field is g(r) = (G * M_enclosed(r))/r², where M_enclosed(r) is the mass within a sphere of radius r.
     M_enclosed(r) = ∫₀ʳ 4πs²ρ(s) ds = ∫₀ʳ 4πs²((ρ₀/R²) * s²) ds = (4πρ₀/R²) * ∫₀ʳ s⁴ ds = (4πρ₀/R²) * (r⁵/5) = (4πρ₀r⁵)/(5R²).
     g(r) = (G/r²) * ((4πρ₀r⁵)/(5R²)) = (4πGρ₀r³)/(5R²) for r ≤ R.
     In terms of total mass M: g(r) = (GMr³)/R⁵ for r ≤ R.
   • Outside the planet (r > R):
     The gravitational field is the same as if the total mass M were concentrated at the center.
     g(r) = GM/r² = (G * ((4πρ₀R³)/5))/r² = (4πGρ₀R³)/(5r²) for r > R.

4. Gravitational potential V(r) at a distance r from the center:

   • Outside the planet (r > R):
     V(r) = -∫∞ʳ g(r') dr' = -∫∞ʳ (GM/r'²) dr' = [GM/r']∞ʳ = GM/r for r > R.
     In terms of ρ₀: V(r) = (G * ((4πρ₀R³)/5))/r = (4πGρ₀R³)/(5r) for r > R.
   • Inside the planet (r ≤ R):
     V(r) = V(R) - ∫ᵣʳ g(r') dr', where V(R) = GM/R.
     V(r) = GM/R - ∫ᵣʳ (GMr'³)/R⁵ dr' = GM/R - (GM/R⁵) * ∫ᵣʳ r'³ dr' = GM/R - (GM/R⁵) * [r'⁴/4]ᵣʳ
     V(r) = GM/R - (GM/R⁵) * ((r⁴/4) - (R⁴/4)) = GM/R - (GMr⁴)/(4R⁵) + (GMR⁴)/(4R⁵) = GM/R - (GMr⁴)/(4R⁵) + (GM)/(4R)
     V(r) = (5GM)/(4R) - (GMr⁴)/(4R⁵) for r ≤ R.
     In terms of ρ₀: V(r) = (5G)/(4R) * ((4πρ₀R³)/5) - (G * r⁴)/(4R⁵) * ((4πρ₀R³)/5) = πGρ₀R² - (πGρ₀r⁴)/(5R²) for r ≤ R.

Since no specific question is asked, we provide the derived properties.