Question

A non-homogeneous sphere of radius R has the following density relation:

$straight rho space equals space straight rho subscript straight o space space space space space space space space space space space space semicolon space 0 space less than space straight r space less or equal than space straight R divided by 2 space space space space equals space 2 straight rho subscript straight o space space space space space space space space space semicolon space straight R over 2 space less than thin space straight r space less or equal than space fraction numerator 3 straight R over denominator 2 end fraction space space space space space equals space 0 space space space space space space space space space space space space space semicolon space space straight r space greater than thin space fraction numerator 3 straight R over denominator 2 end fraction space$

What is the total mass of sphere and gravitational field at the surface of sphere?

Solution

According to the figure given above, we have
Mass of inner sphere of radius R/2 is,

straight m subscript 1 space equals 4 over 3 straight pi open parentheses straight R over 2 close parentheses cubed space straight rho subscript straight o space space space space space space equals fraction numerator pi R cubed straight rho subscript straight o over denominator 6 end fraction

Mass of outer sphere whose outer radius is

fraction numerator 3 straight R over denominator 2 end fraction

and inner

straight R over 2

is,

straight m subscript 2 equals 4 over 3 straight pi open square brackets open parentheses fraction numerator 3 straight R over denominator 2 end fraction close parentheses cubed minus open parentheses straight R over 2 close parentheses cubed close square brackets 2 straight rho subscript straight o

equals 4 over 3 straight pi fraction numerator 26 straight R cubed over denominator 8 end fraction 2 straight rho subscript straight o equals 52 over 6 straight R cubed pi rho subscript straight o

Total mass of the sphere is,

straight m subscript 1 plus straight m subscript 2 space equals fraction numerator pi R cubed straight rho subscript straight o over denominator 6 end fraction plus 52 over 6 pi R cubed straight rho subscript straight o space space space space space space space space space space space space space space space equals 53 over 6 pi R cubed straight rho subscript straight o

Now gravitational field at the surface of sphere is,

space straight E subscript straight g equals fraction numerator G M over denominator straight r squared end fraction space space space space space space equals straight G 53 over 6 fraction numerator pi R cubed straight rho subscript straight o over denominator open parentheses begin display style 3 over 2 straight R end style close parentheses squared end fraction space space space space space space equals 106 over 27 pi R rho subscript straight o straight G

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