A spherical planet of radius R and total mass M has uniform density. Use the shell theorem to derive gravitational-field magnitude g(r) for r R. Show that the field is continuous and maximal at the surface. Derive the small-oscillation period of a test mass moving through a narrow frictionless tunnel along a diameter. Evaluate the period for mean density 5500 kg m⁻³ using G = 6.674 × 10⁻¹¹ N m² kg⁻².

Question

A spherical planet of radius R and total mass M has uniform density. Use the shell theorem to derive gravitational-field magnitude g(r) for r < R and r > R. Show that the field is continuous and maximal at the surface. Derive the small-oscillation period of a test mass moving through a narrow frictionless tunnel along a diameter. Evaluate the period for mean density 5500 kg m⁻³ using G = 6.674 × 10⁻¹¹ N m² kg⁻².

Accepted Answer

Inside g = GMr/R³; outside g = GM/r². Tunnel motion is SHM with ω = √(GM/R³) = √(4πGρ/3), giving T = √(3π/(Gρ)) ≈ 5.07 × 10³ s, or 84.5 min. Uniform density gives enclosed mass M_r = M(r³/R³). Exterior spherical shells exert zero net force on an interior test mass by the shell theorem. Inside, g = GM_r/r² = GMr/R³, directed toward the centre. Outside, the whole sphere acts like a point mass and g = GM/r². At r = R both expressions give GM/R², so the field is continuous. It rises linearly inside and falls as inverse square outside, making the surface value the maximum. Choose signed coordinate x through the tunnel. Acceleration is a = −GMx/R³, the SHM form a = −ω²x, so ω = √(GM/R³) = √(4πGρ/3). The period is T = 2π/ω = √(3π/(Gρ)). For ρ = 5500 kg m⁻³ this is 5.07 × 10³ s = 84.5 min. The ideal result ignores rotation, density variation, air and tunnel engineering.

Field inside and outside a uniform planet

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