Name: How to Escape Gravity
Uploaded: 2019-07-19
Duration: 6 min 57 s
Description: Is there any place in the universe where we would be free of the pull of gravity? If yes, what would we have to do? We are sure you will have an answer after...

Question

Assume earth to be a sphere of uniform mass density. The energy needed to completely disassemble the planet earth against the gravitational pull amongst its constituent particles is

x \times 10^{31} J .

Find the value of

x

. Given the product of mass of earth and radius of earth to be

2.5 \times 10^{31} kg m and g = 10 m s^{- 2} .

Accepted Answer

Given product of mass of earth and radius of earth $= M_{E} R_{E} = 2.5 \times 10^{31} kg m^{- 1}$

Acceleration due to gravity $= g = 10 m s^{- 2}$

Energy needed to disassemble the planet (earth) $= x \times 10^{31} J$

Density of earth $= ρ = \frac{M}{\frac{4}{3} π R^{3}} = \frac{3 M}{4 π R^{3}}$

Let us consider that earth is made by so many spherical shells.so the mass of thickness $d x$

$= d m = 4 π x^{2} d x$ , mass of sphere of radius $x$ , $= m = \frac{4}{3} π x^{3} \cdot ρ$

Now the change in work done $= \frac{G \cdot m \cdot d m}{x} = G (\frac{4}{3} π x^{3} ρ) \cdot \frac{4 π x^{2} ρ d x}{x} = \frac{16}{3} G π^{2} ρ^{2} \cdot x^{4} d x$

$∴$ Total energy needed for disassemble

$= \int_{0}^{R} \frac{16}{3} π^{2} G ρ^{2} \cdot x^{4} d x = \frac{16 π^{2} G ρ^{2}}{3} {[\frac{x^{5}}{5}]}_{0}^{R}$

$= \frac{16}{3} π^{2} G ρ^{2} \cdot \frac{R^{5}}{5} = \frac{16 π^{2}}{3} (\frac{g R^{2}}{M}) {(\frac{3 M}{4 π R^{3}})}^{2} \cdot \frac{R^{5}}{5} = \frac{3}{5} m g R$

where $g = \frac{G M}{R^{2}} \Rightarrow G = \frac{g R^{2}}{M}$

$\Rightarrow E_{T} = \frac{3}{5} m g R = \frac{3}{5} g (M R)$ $= \frac{3}{5} \times 10 \times 2.5 \times 10^{31}$ $= 1.5 \times 10^{31} J$

But in question, $E_{T} = x \times 10^{31}$

$\Rightarrow x \times 10^{31} = 1.5 \times 10^{31} \Rightarrow x = 1.5$

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