Study Guide

Rotation Themes

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  • Classical Mechanics

    The Dark Side of the Moon

    Every time you've ever looked up at a full moon, it's looked just like this:

    This is the only moon that mankind has ever known, the only side of our satellite that ever faces Earth. The moon is stuck in a perpetual staring contest with the Earth due to a phenomenon called tidal locking, where the same forces responsible for tides on Earth have conspired to slow down the moon's rotation until it revolves once around its own axis in exactly the same amount of time it takes the moon to orbit once around the earth.

    This is no cosmic coincidence, but rather the rate of rotation that naturally minimizes the torque on the moon. Minimums, in general, are good—they're why balls roll down hills (minimum potential energy), why planets are spherical (minimum surface area)2, why a quarter pounder is so cheap (minimum wage). Nature likes minimums, and so it's not surprising that she strives for one in satellite orbits, as well.

    How, exactly, nature manages to change the moon's rotational speed is trickier. It's very similar to tides on Earth—the water on Earth directly beneath the moon is pulled towards our satellite by gravity, contributing to"high tide":

    The force of gravity between the Earth and moon is so strong that the moon itself is stretched in much the same way, so that it's not truly a sphere but rather what mathematicians and rugby players call a prolate spheroid (though admittedly just barely).

    Let's rewind the clock about four billion years. (Please don't try this at home. Or if you do, at least wear safety goggles.) The moon, newly formed—or perhaps collected, ejected, trapped, summoned, etc3—spins rapidly around its axis, revolving many times during a single orbit around the earth. It spins so rapidly, in fact, that the deformation caused by Earth's gravity can't keep up with its rotation. The moon starts to bulge, pulled towards the Earth, but rotates quickly away—and the next section of the moon is pulled, but then rotates away, so the next section is pulled, and so on and so on.

    This means the mass of our early, rapidly spinning moon is never evenly distributed. It sticks out to the side like a rocky cowlick, and so the earth's gravity tugs at the moon unevenly, too.

    The uneven force of gravity pulling on this stretched lump of moon rock creates a torque around the moon's axis:

    This torque is directed in the opposite direction of the moon's rotation—gradually slowing the rotation of the moon. In fact, the torque only disappears when the rotational period of the moon about its own axis is exactly the same as its orbital period. When that happens, the prolate spheroid shape of the moon is directed straight along the line between the earth and the moon, and its mass is evenly distributed.

    Once the periods are the same, only one side of the moon ever faces the earth. It wasn't until 1959 that a Soviet lunar probe, Luna 3, took the first recorded images of the dark side of the moon4.

    Spoiler alert—it looks like this:

  • Dynamics

    Spin and Ball Sports

    As anyone who's ever played tennis, soccer, golf, baseball, cricket, polo (normal variety), billiards, dodgeball, polo (elephant variety), ping pong, or polo (Segway variety) can attest, rotation plays a vital roll in almost any sport with a ball. Tennis balls, baseballs, golf balls, or any other kind of roughly spherical object that's thrown, kicked, hit, booted, walloped, or otherwise generally moved through the air from one point to another don't just slide along—they rotate.

    This rotation is called the ball's spin, and it can be in any direction. However, we generally categorize spin into two buckets: topspin and backspin.

    Topspin occurs when the top of the ball rotates forward in the same direction as its travel:

    Backspin, on the other hand, is when the top of the ball rotates backwards, opposite the direction the ball is traveling in:

    The two kinds of spin have very different effects on the path a ball follows. Imagine a soccer player kicking a ball (or a tennis player hitting one, a baseball player throwing one, whatever). With no spin, the trajectory might look something like this:

    Topspin will tend to push the ball down as it travels:

    While backspin will tend to lift the ball slightly:

    The change in trajectory caused by spin is due to something called the Magnus effect.5 A spinning ball that flies through the air will deflect the air behind it as it travels. Topspin tends to lift air up; backspin tends to push air down. Newton's laws tell us that if the air moves, the ball must move in the opposite direction—and sure enough, topspin lifts air up and pushes the ball down, while backspin pushes air down and lifts the ball up.

    This effect can be seen using a wind tunnel, a fog machine, and hopefully lasers because any time you use a fog machine without lasers is a wasted use of a fog machine. In the wind tunnel, a strong wind can be blown over a stationary spinning ball. This both simulates the ball traveling through the air and allows the air deflection to be seen due to the fog:

    In this picture, the wind comes in from the right, simulating a ball traveling to the right. The ball is spinning counterclockwise (backspin), and you can see the air pushed down in its wake. This creates an upward force on the ball, lifting it from its normal trajectory.

    Of course, by the time Gustav Magnus figured this out, athletes had been using the idea to play golf or tennis for (literally) centuries. Whether Magnus ever escaped the lab and tried his theory out for himself on the tennis court is up for debate.

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