g/A skateboarder rises to
the edge of an empty pool and
then falls back down.h/The sum of kinetic plus
gravitational energy is constant.since we don’t know how the person’s metabolism ebbs and flows
over the course of a day. What we can really compute is∆E/∆t,
which is the power averaged over a one-day period.
Converting to joules, we find∆E= 8× 106 J for the amount of
energy transformed into heat within our bodies in one day. Con-
verting the time interval likewise into SI units,∆t = 9× 104 s.
Dividing, we find that our power is 90 J/s = 90 W, about the same
as a lightbulb.2.1.5 Gravitational energy
Gravitational energy, to which I’ve already alluded, is different
from heat and kinetic energy in an important way. Heat and kinetic
energy are properties of a single object, whereas gravitational energy
describes an interaction between two objects. When the skater in
figures g and h is at the top, his distance from the bulk of the planet
earth is greater. Since we observe his kinetic energy decreasing on
the way up, there must be some other form of energy that is increas-
ing. We invent a new form of energy, called gravitational energy, and
writtenU orUg, which depends on the distance between his body
and the planet. Where is this energy? It’s not in the skater’s body,
and it’s not inside the earth, either, since it takes two to tango. If
either object didn’t exist, there wouldn’t be any interaction or any
way to measure a distance, so it wouldn’t make sense to talk about
a distance-dependent energy. Just as marriage is a relationship be-
tween two people, gravitational energy is a relationship between two
objects.
There is no precise way to define the distance between the skater
and the earth, since both are objects that have finite size. As dis-
cussed in more detail in section 2.3, gravity is one of the fundamen-
tal forces of nature, a universal attraction between any two particles
that have mass. Each atom in the skater’s body is at a definite dis-
tance from each atom in the earth, but each of these distances is
different. An atom in his foot is only a few centimeters from some
of the atoms in the plaster side of the pool, but most of the earth’s
atoms are thousands of kilometers away from him. In theory, we
might have to add up the contribution to the gravitational energy
for every interaction between an atom in the skater’s body and an
atom in the earth.
For our present purposes, however, there is a far simpler and
more practical way to solve problems. In any region of the earth’s
surface, there is a direction called “down,” which we can establish
by dropping a rock or hanging a plumb bob. In figure h, the skater is
moving up and down in one dimension, and if we did measurements
of his kinetic energy, like the made-up data in the figure, we could
infer his gravitational energy. As long as we stay within a relatively
small range of heights, we find that an object’s gravitational energy
increases at a steady rate with height. In other words, the strength
Section 2.1 Energy 81