Simple Nature - Light and Matter

f/The gravitational energy
U = −Gm 1 m 2 /r graphed as a
function ofr.

rience with gravity near the earth’s surface thatU is proportional

to the mass of the object that interacts with the earth gravitation-

ally, so it makes sense to assume the relationship is symmetric:Uis

presumably proportional to the productm 1 m 2. We can no longer

assume ∆U∝∆r, as in the earth’s-surface equation ∆U=mg∆y,

since we are trying to construct an equation that would be valid

for all values ofr, andg depends on r. We can, however, con-

sider an infinitesimally small change in distance dr, for which we’ll

have dU =m 2 g 1 dr, whereg 1 is the gravitational field created by

m 1. (We could just as well have written this as dU=m 1 g 2 dr, since

we’re not assuming either mass is “special” or “active.”) Integrating

this equation, we have

∫

dU=

∫

m 2 g 1 dr

U=m 2

∫

g 1 dr

U∝m 1 m 2

∫

1

r^2

dr

U∝−

m 1 m 2

r

,

where we’re free to take the constant of integration to be equal to

zero, since gravitational energy is never a well-defined quantity in

absolute terms. WritingGfor the constant of proportionality, we

have the following fundamental description of gravitational interac-

tions:

U=−

Gm 1 m 2

r

[gravitational energy of two masses

separated by a distancer]

We’ll refer to this as Newton’s law of gravity, although in reality

he stated it in an entirely different form, which turns out to be

mathematically equivalent to this one.

Let’s interpret his result. First, don’t get hung up on the fact

that it’s negative, since it’s only differences in gravitational energy

that have physical significance. The graph in figure f could be shifted

up or down without having any physical effect. The slope of this

graph relates to the strength of the gravitational field. For instance,

suppose figure f is a graph of the gravitational energy of an asteroid

interacting with the sun. If the asteroid drops straight toward the

sun, 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 and

D, 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 gravitational

energy, and therefore speeds up considerably. This is due to the

stronger gravitational field.

100 Chapter 2 Conservation of Energy