Energy and Human Ambitions on a Finite Planet, 2021a

D.4 Pushing Out the Moon 403
aggressive use of tidal energy has the power to push the moon away
from Earth, providing the mechanism by which we could “use up” this
resource. Curious students dem<strong>and</strong>ed an explanation. Even though it’s
not of any practical importance, the physics is neat enough that the
explanation can at least go in an appendix.
less pull
more pull
Moon
(10× farther)
Figure D.1: The moon pulls harder on the
near side of the earth, <strong>and</strong> less hard on the
back side. Relative to the earth as a whole
(medium force), the near side advances toward
the moon <strong>and</strong> the back side lags the
rest of the earth, creating a bulge on both
sides that is aligned toward the moon. Note
that a drawing to scale would put the moon
well off the page.
The first step is realizing that Earth <strong>and</strong> Moon each pull on each other 36 via
gravitation. Since the strength of gravity decreases in proportion to the
square of the distance between objects, the side of the earth closest to the
moon is pulled more strongly than the center of the earth, <strong>and</strong> the side
opposite the moon is pulled less strongly. The result is an elongation of
the earth into a bulge—mostly manifested in the oceans (Figure D.1).
36: In fact, equally, per Newton’s third law.
rotation drags bulge
Moon
Figure D.2: The rotation of Earth <strong>and</strong> it
continents “underneath” the tidal bulge
creates a friction, or drag, that pulls the
bulge around a few degrees (somewhat
exaggerated here), so that it no longer points
directly at the moon.
The second step is to appreciate that the earth rotates “underneath” the
moon, so that the bulge—pointing at the moon—is not locked in place
relative to continents. 37 But friction between l<strong>and</strong> <strong>and</strong> water “drag” the
bulge around, very slightly rotating the bulge to point a little ahead 38 of bulge (high tide) every ∼ 12 hours.
the moon’s position (Figure D.2).
38: The angular shift is around 1–2 ◦ .
37: This is why we experience two high
tides per day <strong>and</strong> two low tides: the earth is
spinning underneath the opposite bulges,
so that a site on the surface passes under a
orbital velocity
resultant (mostly to Earth) has nudge to side
Now think about how the moon sees the earth, gravitationally. It mostly
sees a spherical earth, but also a bulge on the front side, slightly displaced,
<strong>and</strong> a bulge on the back side, also displaced in the opposite direction
(Figure D.3). While the bulge masses are equal, the closer one has a
greater gravitational influence <strong>and</strong> acts to pull the moon a little forward
in its orbit, speeding it up. 39
Figure D.3: Gravitationally, the earth looks
like a big central mass <strong>and</strong> two bulge masses
displaced from the connecting line. The
closer mass pulls harder than the more distant
one, so the addition of all the force
vectors (not to scale) results in a little asymmetry,
leaving a small sideways component
of the force along the same direction as the
moon’s orbital velocity (up in this drawing).
39: It may help to think of this bulge as
being like a carrot dangled in front of a
horse, encouraging it forward.
Accelerating an orbiting object along its trajectory adds energy to the
orbit <strong>and</strong> allows the object to “climb” a little farther away from the
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.