imagine there's a tunnel drilled across the centre of the Earth where a capsule moves under gravity serving as transport between Europe and the Antipodes. can you estimate the travel time? what would be the largest velocity? is there something funny about the result if you assume homogeneity?

the equation of motion in general reads

ReplyDelete\begin{equation}

\ddot{r} = -\frac{G}{r^2}\int_0^r4\pi r^2\mathrm{d}r\:\rho(r)

\end{equation}

which simplifies to $\ddot{r} = -4\pi G/3\:\rho r$ for homogeneous densities. that differential equation describes a harmonic oscillation with the angular frequency $\omega=\sqrt{4\pi/3\:G\rho}$ and the time constant $T=2\pi/\omega\simeq 1.5$ hours (assuming a mean density of 5.000 kilograms per cubic meter). of couse one could have guessed the result noticing that $\sqrt{G\rho}$ defines an inverse time scale by dimensional arguments.

CQW likes to add that for homogeneous densities the travel time does not depend on the size of the planet, which is an expression of a pendulum's isochronism. in terms of mechanical similarity this is written as a relation $r^{2-n}\propto t^2$ in a potential of the shape $\Phi\propto r^n$, which is obtained by the growth of the enclosed mass $\propto r^3$ and the $1/r$-dependence of the Coulomb-potential.

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