Originally posted by debbiejoKewl. I have one in mind. Just give me a few minutes to pack my bags and rent a shuttle.
He wants to blow up a planet. 🙁
Anyway, I found this...
"Calculations:
Notations to be used:
r = distance from core of planet
R = Radius of planet
p = density of planet
G = Newton's gravitational constant = 6.67 x 10^-11
M = mass of core
m = mass of planet = p times V
U = gravitational potential energy
E = energy (total)
pi = 3.14159...etc
V = volume, for a sphere = (4/3)pi R^3
I will calculate the gravitational potential energy between a "shell" of the planet and the core, then integrate from r=0 to r=R. A shell can be imagined by visualizing the planet as composed of many layers of infinetely thin shells with thickness dr. There would be shells at every distance r from 0 to R.
A shell would have mass dm = p times dV. dV is the volume of the shell, which would be the surface area of the shell times the thickness. So dV =
(4*pi*r^2)*dr. Thus, dm = p * (4*pi*r^2)*dr
First, the gravitational potential energy is:
U(r) = -GMm / r
So, the potential energy of a shell of mass dm at distance r would be
dU(r) = -GM dm / r
The mass M is the mass of the core: ie. the mass of everything from radius 0 to radius r where the shell is. From the shell theorem, we can regard this mass as acting as a point mass at the centre of the planet. Mass is p times V so M = p*(4/3)*pi*r^3.
So then plugging in the above values for M and dm, we get:
dU(r) = -(16/3)G*(p^2)*(pi^2)*(r^4)*dr
Everything on the right is a constant except for the (r^4)dr part, whose integral is (1/5)*r^5. Now integrating the above expression from r=0 to r=R, we will get:
E = U (total) = -(16/3)*G*(p^2)*(pi^2)*(R^5)
So then, to free all these particles and explode the planet, we need to make the total energy be at least 0. So if we gave the planet the amount of energy E as above then it would explode. Any extra energy more than E would give the particles kinetic energy and determines how fast the pieces fly about.
For the Earth, R = 6.38x10^6 metres.
And density, under our assumptions = mass / volume = 5497 kg/m^3
Putting these numbers in, we get the energy to "explode" the Earth to be 2 x 10^32 J. So then the magnitude of energy to explode the Earth is somewhere in the range of 10^32 J."
💃