Question

Assume that a Spherical Planet Of Radius R, Has a Uniform Mass Density (Per Unit Volume) Distribution Throughout, Of Value Po. Also, Assume that There Is a Massive Dust Cloud In the Rest Of the Universe, Which Decays Exponentially In Radius, r, Away From the Surface Of the Planet, Where the Mass Density Varies As ρ(r) = Po exp| | | |, For r2R- a) Using the Integral Form Of Gauss's 6. Law, [n.gda--4πGJsoh', And Spherical Coordinates (Specifically Using the Radial Coordinate, r). For the Gravitational Field Vector, g. Determine the Gravitational Field Vector, g().As a Function Of Radius, r, Expressed In Terms Of the Given Parameters, G,R Po, the Radial Unit Vector, F, And An Appropriate Radial Integral, From the Planet Surface Radius、 R,, To a General Radius, r, For r R, Also, Provide the Result, g(r-R).At r - R. b) After Utilizing a Change Of Variables In the Needed Integral, Using An Appropriate Coordinate, Call It x, Re- write the Gravitational Field Expression So That the Integral Goes From x =0 To An Appropriate Function or r. By Utilizing the Factorial Function Infinite Exponential Integral, n|sfdor''eヅ, where 0-1. Determine the Asymptotic, As r oo, Expression For the Gravitational Vector Field, g(r), As a Function Of r . And the Given Parameters, G, R, Po Note That the Exponential Integrals Can Be Evaluated At r-oo, Which Leaves a Typical Gravitational Field Expression, Which Includes the Effective Mass Of the Planet As Well As the Mass Of the Infinite Dust Cloud

Answer 1

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