A solid spherical planet of radius R has density ρ(r) = ρ₀r. Its gravitational acceleration is half of its maximum value at an internal distance x and also at an external distance y from the centre. If x + y = R√(α/β), where α and β are single-digit integers and α/β is in lowest form, find α - β.

Question

G1 Physics · Accepted Answer

Step 1: ρ=ρ₀r givₑ M(r)=4πρ₀∫₀ʳr'³dr'=πρ₀r⁴, hence gᵢₙ=Gπρ₀r². Thus gₘₐₓ=Gπρ₀R² at the surface. Step 2: g(x)=gₘₐₓ/2 givₑ x=R/√2. Outside, g(y)=Gπρ₀R⁴/y²=gₘₐₓ/2 givₑ y=√2R. Step 3: x+y=(3/√2)R=R√(9/2), so α=9, β=2 and α-β=7.