Because the formulas for Coulomb's law and Newton's law of gravity have the same inverse-square law dependence on distance, a formula analogous to the formula for Gauss's law can be found for gravity. The gravitational ficld ĝ at a location is the force per unit mass on a test mass mo placed at that location. (Then, for a point mass m at the origin, the gravitational ficld g at some position 7' is j = – (Gym/r²) î.) (a) Compute the flux of the gravitational ficld through a sphcrical surface of radius r, and verify that the gravitational analog of Gauss's law is dÃ= -4TGNMencl· (b) What do you think is the differential form of Gauss's law for gravitation? (c) Determine the gravitational ficld, g, a distance r from the center of the Earth (where r < RE, where RE is the radius of the Earth), assuming that the Earth's density is uniform. Express your answer in terms of GN, the mass of the Earth, МЕ, and Rg.

Because the formulas for Coulomb's law and Newton's law of gravity have the same inverse-square law dependence on distance, a formula analogous to the formula for Gauss's law can be found for gravity. The gravitational ficld ĝ at a location is the force per unit mass on a test mass mo placed at that location. (Then, for a point mass m at the origin, the gravitational ficld g at some position 7' is j = – (Gym/r²) î.) (a) Compute the flux of the gravitational ficld through a sphcrical surface of radius r, and verify that the gravitational analog of Gauss's law is dÃ= -4TGNMencl· (b) What do you think is the differential form of Gauss's law for gravitation? (c) Determine the gravitational ficld, g, a distance r from the center of the Earth (where r < RE, where RE is the radius of the Earth), assuming that the Earth's density is uniform. Express your answer in terms of GN, the mass of the Earth, МЕ, and Rg.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter10: Motion In A Noninertial Reference Frame

Section: Chapter Questions

Problem 10.13P

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Concept explainers

Ising model of Ferromagnets

The Ising model is one of the easiest and most famous theoretical models for ferromagnetism in statistical mechanics. When a group of atomic spins aligns such that their magnetic moments are pointed in the same direction and produce a strong permanent magnetic moment in macroscopic size, the phenomenon is called ferromagnetism. The relative permeability of these magnets is greater than unity and increasing magnetization with the applied external field. Even in the absence of an external magnetic field, these substances retain spontaneous magnetization. An example of ferromagnets is iron.

Question

Because the formulas for Coulomb's law and Newton's law of gravity have the same
inverse-square law dependence on distance, a formula analogous to the formula for
Gauss's law can be found for gravity. The gravitational ficld g at a location is the force
per unit mass on a test mass mo placed at that location. (Then, for a point mass m
at the origin, the gravitational field g at some position is g = - (GNm/r²)î.)
(a) Compute the flux of the gravitational ficld through a spherical surface of radius
r, and verify that the gravitational analog of Gauss's law is
*, = f i
dÃ = -4TGNMenel·
(b) What do you think is the differential form of Gauss's law for gravitation?
(c) Determine the gravitational field, g, a distance r from the center of the Earth
(where r < RE, where RE is the radius of the Earth), assuming that the Earth's
density is uniform. Express your answer in terms of GN, the mass of the Earth,
МЕ, and Rp-
Using the results of the problem above, suppose that you were to drill a tunnel all
the way through the Earth, and then drop a ball down the tunnel. Neglecting any
friction/air resistance/cffects due to the rotation of the Earth, cxplain what would
happen to the ball and how long you would have to wait (in minutes!) to get it back!
Hint: consider the force on the ball when it is inside the Earth. Does this form of the
force look familiar? This same method also works for a small charge falling through an
oppositely-charged big ball of charge, of course.

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