Goodbye

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This hopefully fills out the details in Gautam's great solution above. First from the problem statement let us convince ourself that it is asking for the potential energy of the system and that this is equivalent to the gravitational binding energy. If we bash it out and write the equation for bringing each infinitesimal mass to infinity, we will discover that $U$ is independent of the order in which the masses were disassembled. Since it is independent also of the route by which the masses were brought, $U$ must be a unique property of the final configuration of the masses. Thus $U$ is just the gravitational potential energy of this system. This can be found by just finding the energy of assembly and taking the negative.

Now for the actual problem, consider the diagram above. We can assemble the planet by bringing in thin shells of mass $dm$ to our partially assembled mass $m$ of radius $r$ . This partial mass will cause a potential $\phi(r) = -\int_{\infty}^r \! \frac{Gm\hat{r} dr}{r^2} = \frac{-Gm}{r}$

The energy required to bring the thin shell from infinity $du$ is

$dU = \phi(r) dm$ We also know that $\rho$ is constant, thus $m = \rho \frac{4\pi r^3}{3}$ and $dm = \rho dV = \rho 4 \pi r^2 dr$

Substituting these two expressions in our first , we get

$dU = \frac{16}{3} G\rho^2 \pi^2 r^4 dr$

Integrating this from $0$ to $R$ and simplifing , we get

$U = \frac{3Gm^2}{5R}$

As an extra fun exercise, think about how we to attack this problem using just dimensional analysis.

"The energy required to irreversibly blow up our lovely planet into smithereens" should be equal to self-gravitation potential energy of sphere because it is written that earth is made of neutral particles. $\text{From energy conservation}$ $\text{Self gravitational potential enrgy+Energy supplied=0}$ At infinity GPE of a body is treated as zero hence it will be irreversible. $\text{Self GPE}=-\frac{3GM^2}{5R}$ (for proof see this )

Hence $\frac{3GM^2}{5R}=energy\space supplied$ substitute values and we get answer $2.241\times 10^{32}$

"The energy required to irreversibly blow up our lovely planet into smithereens" should be equal to self-gravitation potential energy of sphere because it is written that earth is made of neutral particles. $$ We can assemble the planet by bringing in thin shells of mass to our partially assembled mass of radius . This partial mass will cause a potential

The energy is just under the entire output of the Sun in a week. Or the total energy from the Sun arriving at the Earth during about 4 Million Years.

A megaton of TNT yields about 4.184E15 Joules of energy. A thermonuclear warhead is equivalent to about 6 times amount of energy. So, to blow up Earth , we would need about 8.93E15 warheads of that size.