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Frictionless Train Through the Earth

I need some help answering these questions:
a) The gravitational potential energy U(r) of a mass m due to a mass density ρ(r) satisfies ∇^2 U = 4 π Gmρ, where G is the gravitational constant. If the earth is considered to be a uniform sphere of mass M, radius R, show that the gravitational potential energy of a mass m inside the earth a distance r from the center is


where g = GM/R^2 = 9.81 m/s^2.

b) A tunnel is to be constructed through the earth, along which a frictionless train will run between cities at A and B. The track is described by the curve r(ϕ) (polar coordinates). (see attachment for diagram)
Use energy conservation to derive an expression for the speed v of a train which starts at rest at A as it passes through segment dl of the curve at r(ϕ), and hence show that the journey time is.... (see attachment)

where r' = dr/dϕ.

c) The tunnel is to be constructed so as to minimize the journey time. Taking care to note the nature of the integrand, write down an Euler-Lagrange equation for the extremal curve r(ϕ) [not asked to solve this].
Let a be the minimum distance of the tunnel from the center of the earth, where dr/dϕ = 0. Use this to obtain an expression for dr/dϕ in terms of r,R, and a.

d) Noting dϕ = dr/(dr/dϕ), re-express T as an integral (see attachment)

and evaluate.

e) If R = 6,400 km determine the journey time in minutes through the center of the earth.

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Solution Summary

This solution uses conservation energy and momentum to calculate the travel time of a frictionless train through a spherical planet of homogeneous mass density with step-by-step calculations and workings.