Simple Nature - Light and Matter

to the mass of the object that interacts with the earth gravitationally,so it makes sense to assume the relationship is symmetric: U ispresumably proportional to the product m 1 m 2 . We can no longerassume ∆U ∝ ∆r, as in the earth’s-surface equation ∆U = mg∆y,since we are trying to construct an equation that would be validfor all values of r, and g depends on r. We can, however, consideran infinitesimally small change in distance dr, for which we’llhave dU = m 2 g 1 dr, where g 1 is the gravitational field created bym 1 . (We could just as well have written this as dU = m 1 g 2 dr, sincewe’re not assuming either mass is “special” or “active.”) Integratingthis equation, we have∫∫dU = m 2 g 1 dr∫U = m 2 g 1 dr∫ 1U ∝ m 1 m 2r 2 drU ∝ − m 1m 2r,where we’re free to take the constant of integration to be equal tozero, since gravitational energy is never a well-defined quantity inabsolute terms. Writing G for the constant of proportionality, wehave the following fundamental description of gravitational interactions:U = − Gm 1m 2r[gravitational energy of two massesseparated by a distance r]f / The gravitational energyU = −Gm 1 m 2 /r graphed as afunction of r.We’ll refer to this as Newton’s law of gravity, although in realityhe stated it in an entirely different form, which turns out to bemathematically equivalent to this one.Let’s interpret his result. First, don’t get hung up on the factthat it’s negative, since it’s only differences in gravitational energythat have physical significance. The graph in figure f could be shiftedup or down without having any physical effect. The slope of thisgraph relates to the strength of the gravitational field. For instance,suppose figure f is a graph of the gravitational energy of an asteroidinteracting with the sun. If the asteroid drops straight toward thesun, from A to B, the decrease in gravitational energy is very small,so it won’t speed up very much during that motion. Points C andD, however, are in a region where the graph’s slope is much greater.As the asteroid moves from C to D, it loses a lot of gravitationalenergy, and therefore speeds up considerably. This is due to thestronger gravitational field.100 Chapter 2 Conservation of Energy