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To apply this relationship to the problem of finding the gravitational binding energy of the Earth, we imagine removing one infinitesimally thick outer shell after the other from a spherical distribution of mass that gets smaller and smaller as shells are removed. Each mass shell is moved an infinite distance away through the diminishing force of gravity between the shell and sphere. It will take some energy to move all these shells; that energy is the negative of the binding energy of the system. To find energy needed to remove all these shells, we will find the change in the system's potential energy. To find the change in potential energy, we need to know the mass of the shell, the mass of the remainder of the sphere, and the force through which we're moving the shell.

The mass of any volume is simply density times volume. Each shell has a volume given by its area times its thickness and a mass given by its volume times its density. In short, we have


where is the radius of the sphere.

If we assume that the density of the sphere is the same everywhere, then each shell contributes an infinitesimal portion of potential energy as described by

where the average density is given by

where the angle brackets indicate an average value and

Substituting, we obtain

To find the total change in potential energy as all the shells are removed by summing all the infinitesimal bits of potential energy, we integrate over the radius of all the shells, from the outside to the inside, like this

Thus for any point within the sphere, the change in potential energy using average density is

Therefore the total change in potential energy as the Earth is disassembled using average density becomes

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One source tells us that among 16th century armies the adoption of the two-handers was very limited and in comparison with the pike or the halberd did not play a meaningful role. “In the infantry unit, the German and Swiss Landsknechts positioned the Doppelsöldner (Soldiers who received double pay for wielding the two-handers) in the front ranks for a long time to strike down the opposing pikes and to hack out breaches into which one's own soldiers could penetrate. However it would become unusable, as soon as the opposing forces collided with one another, and there would be increased pressure from the back ranks onto the front ranks, which created a thick melee.” Thus, “sometime around the middle of the 16th century it (the two-hander) disappeared from war and mutated into a form of guard and ceremonial weapon with a symbolic character.” (Kamniker and Krenn, p. 130).

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