Frictionless Train Through the Earth
Not what you're looking for?
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
U(r)=mg/2R(r^2-3R^2)
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.
Purchase this Solution
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.
Purchase this Solution
Free BrainMass Quizzes
Introduction to Nanotechnology/Nanomaterials
This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.
Basic Physics
This quiz will test your knowledge about basic Physics.
Classical Mechanics
This quiz is designed to test and improve your knowledge on Classical Mechanics.
Intro to the Physics Waves
Some short-answer questions involving the basic vocabulary of string, sound, and water waves.
The Moon
Test your knowledge of moon phases and movement.