Finalize Consider the limit on increasing the gravitational force by bringing th

Question

Finalize Consider the limit on increasing the gravitational force by bringing the masses closer together. The two spheres of osmium cannot have their centers closer than the sum of their radii, and if the radii are made smaller, the masses decrease. To what power of the radius r is the mass of each sphere proportional? In proportion to what power of r does the smallest separation of the centers vary? So the attraction between the spheres of maximum mass at smallest separation increases as r to what power?

Explanation / Answer

density = mass / volume

volume = 4 * pi * r^3 / 3

let density be d

then mass = d * volume

mass = d * 4 * pi * r^3 / 3

mass is proportional to the power of r = 3

force of gravitation = k * m^2 / (2r)^2

force of gravitation = k * (d * 4 * pi * r^3 / 3)^2 / (2r)^2

force of gravitation = k * (d * 4 * pi * r^3)^2 / (6r)^2

force of gravitation = k * (d * 4 * pi)^2 * r^4 / 36

power of r = 4

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