Can we slow down the world?

Every now and again, I get a question, either by email, or in this case, via the facebook fanpage that’s fun, but not really appropriate for my column. Facebook and io9 mainstay Richard Henretta who asks:

In the last 100 years, we have been building upwards with ever increasingly tall skyscrapers. As we do so, we are moving the mass of the earth outwards from the center more and more. Keeping with the law of conservation of angular momentum, this would mean the length of our days must be increasing, but by how much?

I’ve put a “technical” tag on this one, because I’m going to be using a few equations, but nothing that you wouldn’t have seen if you took a freshman physics class. Basically, this is going to just be a matter of angular momentum.

I’m going to make a couple of assumptions that may only be approximately true.

1. We’ll pretend the earth is a solid sphere of radius 6400km and mass, $6times 10^{24}kg$, which means that the moment of inertia is:
$I=frac{2}{5} M_oplus R_oplus^2simeq 9.83 10^{37} kgcdot m^2$
If course, the earth doesn’t have a constant density; it’s denser in the center than at the edges. Still, it gives us a decent enough estimate, and the error is probably smaller than the other errors we’re likely to make here.

2. We’re assuming that all of the materials used to build the buildings are drawn from ground level at the equator. Obviously, if they’re mined from deep underground, the effect of increased moment of inertia will be greater. If they’re taken from a mountain top, the net effect might be a decrease in moment of inertia.
3. We assume that the center of mass of the buildings are about halfway up. In reality, this, too, is almost certainly flawed. Buildings tend to be larger at their base than at the top, but again, this is probably good enough for a decent estimate.

The calculation is easy enough. What happens if you take a mass, $m$, and raise it a height, $h$, above sea level (at the equator)? The moment of inertia added to the earth is:
$Delta I=2mR_oplus h$
The Burj Khalifa (pictured above) is currently the tallest building in the world, and is about 828 meters tall, and an empty weight of about 500,000 tons.

Moreover, at a latitude of about 25 degrees, it’s close enough to the equator as to not screw up our calculation too much. Thus:
$Delta I=2.65times 10^{18} kgcdot m^2$
Or in other words, the moment of inertia of the earth increases by:
$frac{Delta I}{I}simeq 2.7times 10^{-20}$
Since moment of inertia is conserved:
$frac{day_{new}}{day_{old}}simeq 1+2.7times 10^{-20}$
In other words under our assumptions, the Burj Khalifa extends a day by about $2.3times 10^{-15} seconds$.

Of course, this effect is cumulative, and since the effect includes both the mass and the height, the effect from lots of midsized buildings almost certainly overwhelms the few at the top.

Still, the man-made effect is tiny compared to that of the moon. Every year, the tidal force from the moon slows down the earth by about 23 microseconds, about 10 billion times that from the Burj Khalifa.

-Dave

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One Response to Can we slow down the world?

1. Richard says:

Math: the more you know!