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There’s nothing you and I won’t do

December 19, 2025 at 9:43 PM by Dr. Drang

I’ve talked about area moments of inertia in the past couple of posts. Today, thanks to Phil Plait, resident astronomer at Scientific American, I have an excuse to talk about mass moments of inertia, the kind you learned about in first semester physics class.

Plait’s article today takes on the Modern English song “I Melt with You” and looks into the energy it would take to stop the world.1 By “stopping,” he initially assumes that means stopping the rotation of the Earth about its axis. He gives the value, 2×1029joules, but doesn’t explain how he calculated that number. Kind of disappointing for SciAm and definitely not acceptable here at ANIAT. Especially since the calculation is easy.

The kinetic energy of a rotating body is

12Iω2

where I is the mass moment of inertia of the body about its axis of rotation and ω is its rotational speed.

The value of I is kind of tricky because, as Plait points out, the Earth’s density varies with depth. But we can make an initial estimate by assuming the Earth is a sphere of uniform density. The moment of inertia for that kind of body is

I=25mr2

We get the mass and (average) radius of the Earth from the sadly defunct NASA Earth Fact Sheet, currently available through the Internet Archive. The mass is 5.9722×1024kg and the mean radius is 6371km. That gives us a moment of inertia of

I=25(5.9722×1024kg)(6.371×106m)2=9.6964×1037kgm2

Relative to the stars, the Earth rotates once every 23.9345 hours (that’s the “sidereal rotation period” on the Fact Sheet), so the rotational speed is

ω=2π(23.9345×60×60)s=7.29211×10−5s−1

Putting these together, we get a kinetic energy of

12(9.6964×1037kgm2)(7.29211×10−5s−1)2=2.578×1029J

This is higher than Plait’s value, but he may have rounded down because he knew the density of the Earth is higher near the center. Also, I suspect he was mostly after an order-of-magnitude answer.

I found a better estimate of I on this page from Eric Weisstein’s World of Science site. It was determined by Kurt Lambeck to be

I=8.034×1037kgm2

(Lambeck’s book came out just a couple of years before “I Melt with You,” which I’m pretty sure is a coincidence.)

Using this value, we get a rotational kinetic energy of

12(8.034×1037kgm2)(7.29211×10−5s−1)2=2.136×1029J

which is a better match for Plait’s energy value.

Plait also considers another interpretation of “I’ll stop the world”: stopping the Earth in its orbit. I won’t get into the details of this, first because it doesn’t involve the moment of inertia, and second because stopping the Earth this way would have it fall into the Sun. Not much doubt about melting in that case.

  1. He also gets into the question of melting the Earth’s crust, which I won’t be addressing. After all, the song says “I’ll stop the world and melt with you.” As far as I’m concerned, only the singer and his girl melt, not the whole goddamned world. ↩

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