If one wants to remove all the mass of the earth to infinity in order to break it up completely.
The amount of energy that needs to be supplied will be x/5GM²/R where x is __________ (Round off to the Nearest Integer) (M is the mass of earth, R is the radius of earth, G is the gravitational constant)

Solution

Physics sir · 4 steps

Build the earth one thin shell at a time; the energy to pull it apart is that same energy turned around.

1. What we need. Scattering every bit of the earth to infinity takes energy equal to the size of its gravitational self-energy U. So find U for a uniform sphere of mass M and radius R.
2. Add one shell. Suppose the sphere has grown to radius r. Its mass is m = Mr³/R³ and the potential at its surface is -Gm/r. Bring in a shell of mass dm = 3Mr²/R³ dr from infinity, and the energy changes by dU = -(Gm dm)/r = -3GM²/R⁶r⁴ dr.

   U = -3GM²/R⁶∫₀^R r⁴ dr = -3GM²/R⁶·R⁵/5 = -3/5GM²/R

3. Energy to supply. To break the earth up you must supply +3/5GM²/R. That is the energy that brings U back up to zero.
4. So. Comparing with x/5GM²/R gives x = 3.

The common slip: 5. Treating the earth like a point mass M sitting at distance R from another mass M gives GM²/R, so x = 5. The mass is spread out, and the shells added early feel a weaker pull.

The idea to keep. A uniform solid sphere has self-energy -3/5GM²/R. A thin shell has -1/2GM²/R. To break up a body completely, supply the size of its self-energy.

Answer from the NTA final answer key. Solution by PrepWiser.