a_{y} = \frac{d^{2}y}{dt^{2}} =

y(t) = A cos (ωt + ϕ)

x(t) = A cos (ωt)

M = \frac{4}{3}πR_{E}^{3} ρ

a = \frac{GM}{R_{E}^{2}}  = \frac{4πGρR_{E}^{3}}{3R_{E}^{2}}  = \frac{4}{3}πGρ R_{E}

m = \frac{4}{3}πx^{3} ρ

a = \frac{Gm}{x^{2}}  = \frac{4πGρx^{3}}{3x^{2}}  = \frac{4}{3}πGρ x

a⃗ = − (\frac{4}{3}πGρ) x⃗

\frac{dx^{2}}{dt^{2}} = − (\frac{4}{3}πGρ) x⃗

ω ≡ \sqrt{\frac{4}{3}πGρ}

x(t) = R_{E} cos(ωt)