g/A view of the earth-moon
system from above the north
pole. All distances have been
highly distorted for legibility.Earth’s slowing rotation and the receding moon example 2
The earth’s rotation is actually slowing down very gradually, with
the kinetic energy being dissipated as heat by friction between the
land and the tidal bulges raised in the seas by the earth’s gravity.
Does this mean that angular momentum is not really perfectly
conserved? No, it just means that the earth is not quite a closed
system by itself. If we consider the earth and moon as a system,
then the angular momentum lost by the earth must be gained by
the moon somehow. In fact very precise measurements of the
distance between the earth and the moon have been carried out
by bouncing laser beams off of a mirror left there by astronauts,
and these measurements show that the moon is receding from
the earth at a rate of 4 centimeters per year! The moon’s greater
value ofrmeans that it has a greater angular momentum, and
the increase turns out to be exactly the amount lost by the earth.
In the days of the dinosaurs, the days were significantly shorter,
and the moon was closer and appeared bigger in the sky.
But what force is causing the moon to speed up, drawing it out
into a larger orbit? It is the gravitational forces of the earth’s tidal
bulges. In figure g, the earth’s rotation is counterclockwise (ar-
row). The moon’s gravity creates a bulge on the side near it, be-
cause its gravitational pull is stronger there, and an “anti-bulge”
on the far side, since its gravity there is weaker. For simplicity, let’s
focus on the tidal bulge closer to the moon. Its frictional force is
trying to slow down the earth’s rotation, so its force on the earth’s
solid crust is toward the bottom of the figure. By Newton’s third
law, the crust must thus make a force on the bulge which is to-
ward the top of the figure. This causes the bulge to be pulled
forward at a slight angle, and the bulge’s gravity therefore pulls
the moon forward, accelerating its orbital motion about the earth
and flinging it outward.
The result would obviously be extremely difficult to calculate di-
rectly, and this is one of those situations where a conservation
law allows us to make precise quantitative statements about the
outcome of a process when the calculation of the process itself
would be prohibitively complex.Restriction to rotation in a plane
Is angular momentum a vector, or a scalar? It does have a di-
rection in space, but it’s a direction of rotation, not a straight-line
direction like the directions of vectors such as velocity or force. It
turns out that there is a way of defining angular momentum as a
vector, but in this section the examples will be confined to a single
plane of rotation, i.e., effectively two-dimensional situations. In this
special case, we can choose to visualize the plane of rotation from
one side or the other, and to define clockwise and counterclockwise
rotation as having opposite signs of angular momentum. “Effec-
Section 4.1 Angular momentum in two dimensions 255