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 sphere centers vary?
So the attraction between the spheres of maximum mass at smallest separation increases as r to what power?
To determine the power of the radius (r) to which the mass of each sphere is proportional, we need to first understand the relationship between mass and radius in the context of the given situation.
Given that the two spheres of osmium cannot have their centers closer than the sum of their radii, we can assume that the smallest separation of the sphere centers is equal to the sum of their radii (2r, where r is the radius of each sphere).
In general, the volume of a sphere is proportional to the cube of its radius (V ∝ r^3). Since the density of osmium is constant, the mass of each sphere is directly proportional to its volume. Therefore, the mass of each sphere is proportional to the cube of its radius (m ∝ r^3).
So, the mass of each sphere is proportional to the cube of its radius: m ∝ r^3.
Now let's consider the smallest separation of the sphere centers. We've established that it is equal to the sum of their radii (2r). In this case, the smallest separation of the sphere centers is directly proportional to the radius (d ∝ r).
Therefore, the smallest separation of the sphere centers varies proportionally to the first power of the radius: d ∝ r^1.
Finally, let's analyze the attraction between the spheres of maximum mass at the smallest separation. According to Newton's law of universal gravitation, the gravitational force between two objects is inversely proportional to the square of the distance between their centers.
So, in this case, the attraction between the spheres varies inversely with the square of the smallest separation of their centers (F ∝ 1/d^2). Substituting d ∝ r, we find that the attraction between the spheres varies inversely with the square of the radius (F ∝ 1/r^2).
Therefore, the attraction between the spheres of maximum mass at the smallest separation increases as the radius (r) decreases to the power of -2.
force= GMm/r^2 but you want m=kr^3 and M = kr^3 because mass is proportional to volume (assuming the radi is the same on each sphere).
force=Gk^2(r^6/r^2)=Constant*r^4