Mass `M`, of a planet earth is uniformly distributed over a spherical volume of radius `R`. Calculate the energy needed to deassemble the planet against the gravitational pull ammongst its consituent particles. Given
`mR = 2.5 xx 10^(31) kg m` and `g = 10 ks^(-2)`.

Explanation

Density of earth, `rho = (M)/((4//3) pi R^(3))`.
Earth is supposed to be made up of a large number of thin concentric spherical sheels. Let the earth be deassembled by removing such shells. Consider a sherical shell of thickness `dx` of earth of radius `x`.
Mass of this sphere of earth, `m_(1) = (4)/(3) pi x^(3) rho`
Mass of spherical shell of radius `x` and thickness
`dx` is, `m_(2) = 4 pi x^(2) dx rho`
The energy needed to deassemble a spherical shell of thichness `dx` from a sphere of radius `x` is
`dW = (Gm_(1)m_(2))/(x) = (G((4)/(3) pi x^(3) rho) (4pi x^(2) dx rho))/(x)`
`= (16)/(3) pi^(2) G rho^(2) x^(4) dx`
Total energy required is
`W =int_(0)^(R) (16)/(3) pi^(2) G rho^(2) x^(4) dx = (16)/(3) pi^(2) G rho^(2) ((x^(5))/(5))_(0)^(R )`
`= (16)/(3) pi^(2) G ((3M)/(4piR^(3)))^(2) ((R^(5))/(5)) = (3 GM^(2))/(4R)`
`= (3)/(5) ((gR^(2))/(M)) xx (M^(2))/(R ) = (3)/(5) g (MR)`
`[ :' g = (GM)/(R^(2))` or `1G = (gR^(2))/(M)]`
`:. W = (2)/(5) xx 10xx (2.5 xx 10^(31)) = 1.5 xx 10^(32) J`