a) The gravitational potential energy U(r)of a mass mdue to a mass density ρ(r)satisfies ∇^2 U= 4πGmρ, whereGis the gravitational constant. If the earth is considered to be a uniform sphere of mass M, radius R, show that thegravitational potential energy of a mass minsidethe earth a distance rfrom the center is

U(r)=mg/2R(r²-3R²)

whereg= GM/R²= 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).

Use energy conservation to derive an expression for the speed v of a train which starts at rest at A as it passes throughsegment dl of the curve at r(Φ), and hence show that the journey time is

T = ∫dt = ∫dl/v = (√R/g _ΦA∫^Φg √(r2+r'2)/R2-r2) dΦ.

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 α be the minimum distance of the tunnel from the center of the earth, where dr/dΦ = 0. Use this to obtain anexpression for dr/dΦ in terms of r,R, and α.

d) Notingd? = dr/(dr/dΦ), reexpressT as an integral

T = 2_a∫^R..dr

and evaluate.

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