How Much is this Girl Slowing Down Earth’s Rotation?
xkcd’s Angular Momentum is a romantic application of the conservation of momentum. But it may lead you to wonder: How does a person spinning counterclockwise cause Earth to slow down? And by how much?
The Bad News: It Probably Doesn’t Slow Down At All
The girl uses the friction between her feet and the ground to spin. Newton’s third law requires that the world “pushes” back, causing it to spin equally in the opposite direction but at a speed that is (roughly) proportional to their masses. This would be a very, very small change in earth’s rotation — but still a change!
The energy from the girl’s push gets transferred to the earth. This is done through friction between every particle between her foot and the effective center of mass of the planet. By the time that energy is transferred, it’s most likely going through so many exchanges that it would have been dissipated as heat before any change in the earth’s rotation is noticed.
So regardless of the lover’s efforts, she’s not doing anything other than getting a workout.
Assuming it Does Slow Down, by How Much Does it Slow Down?
Let’s play along and say all her energy is actually transferred to the earth’s rotation. This means that system of both the earth’s and the girl’s angular momentum L, must be conserved before and after she starts spinning
Angular momentum is nothing more than a constant of proportionality of an object’s moment of inertial I, and its angular velocity ω,
The moment of inertia of an object is just a mathematical representation of how hard it is to spin something about an axis — it depends on the shape and the mass of an object, and the axis on which you’re trying to spin it. For the earth (assuming it’s of uniform density throughout) with radius R and mass M,
The angular momentum of a spinning human body is a little harder determine, but given the mass difference between the earth and a human, we’re pretty safe to model the spinning girl as a cylinder rotating around its symmetry axis with radius R and mass M,
Since angular momentum is conserved, L remains constant, and is equal to the ratio of the moment of inertia of both the earth and girl (which we can approximate to be a uniform sphere due to size differences),
and the angular velocity is
Substituting our moments of inertia into (1) and (2), we arrive at an expression with one unknown quantity: the speed of the slowed down earth with the girl spinning ω,
Location, Location, Location
There’s one last tweak to this relation. The equation above is perfect if the angular momentum vectors for both the girl and the earth are aligned. In other words, if the girl is spinning at precisely the north pole.
The further she gets from the north pole in the northern hemisphere, the less of an effect she has on slowing down the day. This is because only the component of her angular momentum that is parallel to the angular momentum of the earth has an effect on slowing down the day. If she’s at the equator, spinning doesn’t slow down the day at all.
Capturing this effect is an easy fix. After applying some geometry, (3) becomes
Where θ is the degree of latitude of the location of the spinning lover.
Crunching the Numbers
Now it’s time to finally get a number. Applying the following assumptions for known quantities:
Mass of the Earth and system:
Mass of the girl:
Radius of Earth and system:
Radius of the girl:
Angular velocity of system:
Angular velocity of the girl:
Latitude (xkcd is based in Massachusetts):
Throwing these into (4) and solving for the angular velocity for ω after she starts spinning, and recognizing that ω=2π/T, where T is the time in seconds it takes to make a full rotation (one day). We find the difference is approximately
Needless to say, this isn’t very much time gained. But it reinforces the message of comic: any amount of additional time spent with the people you love is worth it.
