Built on Facts

Slowing the earth

Consider the turntable of an old record player. Or equivalently, a CD affixed to a player so that it may spin freely. We’ll pretend there’s no friction, though as always reality will manage to generate some. Now stretch your imagination a bit further and imagine that you shrink yourself down to a height of about 6 inches or so and stand on the turntable/CD. You look around, and having decided that now’s as good a time as any to get started on your exercise routine, you start walking laps around the disc.

Conserving angular momentum, the surface will begin to rotate under you in the opposite direction – though not in general at the same speed. From the viewpoint considering forces, the forward force the floor applies to your foot is matched by the backwards force your foot applies to the floor. When you stop, essentially the same thing happens in the opposite direction.

I’m deliberately skipping all the mathematics here. This is to build intuition.

Building on this force approach to angular momentum, is the following XKCD scheme viable?

i-002d4059f4610e69665e77a05380d69b-xk.png

I am more than certain that this has been beaten to death elsewhere on the web, as Randall Monroe has one of the most erudite fanbases on the internet and probably himself has at least three 9s in his intelligence percentile. Nonetheless, I’d like to give y’all a shot at it. Even the triple 9s can make mistakes…

Comments

  1. #1 Tom
    January 28, 2009

    Yes, the rotation slows down (assuming she’s not at the equator) but when she stops, things go back to normal.

    Spinning isn’t going to be as effective as running east or west, where you can leverage the radius of the earth, but it would take the adult population of the US driving cars east or west at high speed to begin to hope to notice the glitch in the rotation rate of the earth.

  2. #2 DG
    January 28, 2009

    I think it depends on how the atmosphere interacts with the surface. This is what I’m thinking:

    Ignoring air resistance, if she started herself spinning and then somehow levitated off into space, she would have robbed the planet of however much angular momentum she had at that point.

    If instead of levitating off into space, she got herself spinning, spun around once, and then landed back down, she would transfer the angular momentum back to the planet.

    But we have to think about why, if she wants to spin around multiple times, she has to keep pushing off over and over again, rather than just starting herself going once and then somehow keeping herself in the air without transferring any angular momentum (like using a turntable).

    At least part of that has to do with air resistance. If the air were to slow her down without her interacting with the planet at all, she would have to be transferring some of her angular momentum to the air. This is the planet’s missing angular momentum. So, the question is, can the planet recover the angular momentum that’s transferred to the air, or is it lost into an “angular momentum sink” atmosphere forever?

    What do you think?

  3. #3 Blaise Pascal
    January 28, 2009

    I sort of agree with Tom@#1. There are some subtleties where I don’t know if we differ.

    Yes, her spinning will “rob” the Earth of angular momentum. But only while she’s spinning faster and faster. When she spins at a constant rate, so will the Earth (modulo all other effects on the Earth’s spin). When she slows down her spin, she’ll return the “stolen” angular momentum to the Earth.

    Of course, she can’t spin fast enough to really justify the effort. The mass of the Earth is so great that the effect of her spinning at human speeds is going to be immeasurable, and she’ll have wasted the time spinning instead of being there with him. I don’t know what

    But, assuming that we allow them to exceed human capabilities, it’s likely that she’d have to spin fast enough to have relativistic effects — time dilation, etc. I don’t know how that effects spinning objects, but it could be really counterproductive if in order to delay dawn she had to spin so fast as to be the “traveling twin” in the Twin Paradox.

  4. #4 Uncle Al
    January 28, 2009

    1) She should spin at one of the poles but definitely not near the equator.
    2) When hundreds of gigawatts of wind generators are installed and running (all spinning in the same direction because engineering is all right-handed screws)…
    3) There is no time dilation at the rim of a centrifuge,

    http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/mossb.html
    http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/mossfe.html
    then
    Phys. Rev. 129(6) 2371 (1963)
    transverse Doppler effect in an ultracentrufuge hub vs. rim is inert toward rate of time.

    plus

    http://www.geocities.com/physics_world/sr/ae_1905_error.htm
    http://www.physics.gatech.edu/people/faculty/finkelstein/relativity.pdf
    Longitudinal and transverse mass

  5. #5 Matthias
    January 28, 2009

    The spinning slows down the earth, as long as she keeps it up. It doesn’t matter whether she touches the ground or not.

    The slower speed will accumulate, thus delaying the phase of the earth’s rotation permanently. She does lengthen the night, the longer she spins the more so.

  6. #6 dean
    January 28, 2009

    Question:You say “Consider the turntable of an old record player. Or equivalently, a CD affixed to a player so that it may spin freely. We’ll pretend there’s no friction, …”

    If there is no friction, how can your little feet cause the turntable to move in the opposite direction?

  7. #7 rhett
    January 28, 2009

    Tom beat me to this comment. “Each turn” does not slow down the rotation of the Earth. Starting to spin slows down the Earth and stopping spinnning speeds it back up.

    I saw something similar on a TV-stunt show. They were talking about a motorcycle in mid-air changing orientation by speeding up the back wheel. The announcer said “he increases his angular momentum by speeding up” or something like that. In fact, his angular momentum is essentially constant because there are no external torques.

  8. #8 Matthias
    January 28, 2009

    Actually doing the math of what I proposed in #4 the lengthening of the night actually is proportional to the number of turns the girl does.

    With the Moment of Inertia of the Earth Ie and the Girl Ig the total angular momentum L = Ie phi’e + Ig phi’g = const., where phi’e and phi’g are the respective angular velocities, i.e. the time derivatives of phie and phig.

    Thus phi’e = (L – Ig phi’g)/Ie, and phie = int(0,T,phi’e,t) = T L / Ie – T (Ig/Ie) phig.

    Thus the angle the earth spins in a given period of time T is lowered proportionally to the angle the girl spins.

  9. #9 Paul
    January 28, 2009

    No I would say her spinning would not slow down the earth. If all her mass is centered perpendicular to the earth surface, spinning about her own axis would cancel out, so the earth would not “feel” any of the effects.

    I do agree with Tom though. I believe there would be a slowing down if she were to start running opposite the direction of earths rotation. The force she would exert on the surface of the earth would have to be equally opposed and this would be similar (I think) to the turntable example Matt gave.

    Or if running doesn’t fancy you, remember in Superman, when he flew continuously around the earth until the rotation stopped and reversed? I would have to say that that reversal was due to the incredible air resistance he created via his cape (as his suit was so carefully aerodynamic.)

  10. #10 Matthias
    January 28, 2009

    Paul: It’s not the spinning itself that does the trick, it is her pushing so she does begin turning. The bad thing is that she has to stop sometime, and for that she has to push in the opposite direction (friction will do that for her).

    You can’t learn physics from Superman.

    And concerning the Math: the T in the last term should not be there. It’s phie = T L / Ie – phig Ig / Ie.

    Assuming a cylindrical girl of 60kg and a radius of 0.25m for every turn the earth’s rotation is delayed by approx. 5.5*10^-36 degrees. That is the equivalent of 1.33*10^-33 seconds. Not very efficient.

  11. #11 Matthias
    January 28, 2009

    Oh, and I assumed her standing on a pole. Any other location would introduce a factor of sin(latitude), if she stays in the locally upright position. If she stays in parallel with the earth’s axis of rotation everything is as described above, although she might have a hard time completing much more than a full rotation.

  12. #12 Matt Springer
    January 28, 2009

    “If there is no friction, how can your little feet cause the turntable to move in the opposite direction?”

    I mean no friction in the axle. Friction on the surface itself is fine – in fact required for the shoes to grip.

  13. #13 Brian Utterback
    January 28, 2009

    Of course it is viable. It is a romantic gesture and probably worked quite well in that regard. Slowing the Earth, not so much.

  14. #14 dean
    January 28, 2009

    “”If there is no friction, how can your little feet cause the turntable to move in the opposite direction?”

    I mean no friction in the axle. Friction on the surface itself is fine – in fact required for the shoes to grip.”

    I realized friction would be needed on the surface – hence my question. :-) I just missed the description of the location of absence of friction. Sorry.

  15. #15 Peter
    January 28, 2009

    If the planet were rotating in isolation in space then no mater what the young lady does, the total angular momentum will be constant and conserved. Any jumps,twists, running or other more useful activities will at best have miniscule effect on the rotation of the much more massive planet, and depending on what she does, even those are temporary if she returns to her initial conditions relative to the rotating frame of reference attached to the planet.

    But if she permanently moves from her starting latitude to the pole (or equator) the fraction of angular momentum in the system due to her mass rotation about the earth’s axis will decrease (or increase) respectively causing a counter change in the rotational period of the Earth. Alternately if she gets into a rocket ship and goes into an orbit about the Earth she can also remove or add to the rotational energy of the planet depending on the direction and altitude at which she is orbiting.

    Now the Earth is not in pure isolation in space so perhaps there are some second order interactions with the moon and sun which could transfer angular momentum to or from these bodies. This would be effected by using the gravitational forces between her and these bodies to apply torque to the Earth (tidal forces). I believe a sufficiently large displacement in her postion (roughly half the circumference), continuously repeated in synchronization with the varying proximity to the moon or sun due to the rotation of the earth could possibly act, as do the tides, to apply torque to slow the Earth’s rotation. For the ocean tides this is like a second per day per century change – her mass being but a tiny fraction of the ocean’s this is definitely going to be small potatoes.

    Meanwhile continents drifting about, volcanoes bringing up denser material to the surface and movement of material in the molten core give much larger effects that cause observable fluctuations (noise) in the tidal slowing.

  16. #16 CCPhysicist
    January 28, 2009

    Little known “fact”:

    They race cars CCW in NASCAR and CW in F1 … so the perturbation of the earth’s rotation will cancel out. F1 only messes this up with some CW races in the southern hemisphere.
    ;-)

  17. #17 John
    January 29, 2009

    Precession and the attendant torque are taking our mistress away at the rate of about an inch per solar year. Thanks for sparing me the math, Mr. brilliant. Whatever would I have done with it? Good luck in the job market.

  18. #18 rob
    January 30, 2009

    how about this one:

    all the trees on earth suddenly fall over. does the earth speed up or slow down and by what fraction is the change?

    (to get the fraction you need to estimate the total mass of the trees and their height to estimate their contribution to the earths moment of inertia.)

  19. #19 Wayne
    January 30, 2009

    @17: It will speed up, but I don’t have the patience today to estimate the change.

    New theme for the green movement: irresponsible logging is stealing precious moments from your life. ;-)

  20. #20 Tom
    January 31, 2009

    There’s a subtlety to be noted here, that also comes into play in the explanation of leap second. While she turns, the earth’s rotation rate is reduced (by a miniscule amount) but that rate does not keep decreasing. However, since the earth is now “running slow” it will continue to accumulate a phase difference with respect to an independent clock. The longer this happens, the more accumulated time difference there will be.

    It should be noted that this same idea, with regard to things like weather systems, do indeed have an effect on the rotation rate of the earth.

    Another effect one could discuss is simply changing the mass distribution by having people go to the top of a skyscraper, and changing the moment of inertia of the planet.

  21. #21 Johan
    January 31, 2009

    Even if she returns the momentum as she stops spinning the earth moved slower during her spin. This means that a full revolution of the earth would take longer and the night would in fact become longer than it would have been otherwise. Even if the effect is temporary the dawn is still delayed. Or am I thinking about this the wrong way?

  22. #22 JM
    March 17, 2009

    All this talk of small mass.

    Guys, what if she weighs as much as the moon?

  23. #23 raj
    February 8, 2010

    so if she spun clockwise… well,, can she speed up the rotation of earth?

  24. #24 Erika
    February 11, 2010

    Hey guys.., I have a ques. So what will happen to us when the Earth Slows down? If you don’t mind kindly visit our website too! Thank you and God Bless you all! :D

The site is currently under maintenance and will be back shortly. New comments have been disabled during this time, please check back soon.