I’d like to make myself believe
That planet Earth turns slowly…
– Owl City, Fireflies
If you’re had any exposure to pop radio over the last few months, you’ve heard this plaintive rumination about the earth’s rotation. The first time I heard it I thought it must be the Postal Service getting back together, but alas it was not so. Oh well, at least we can get some physics out of it.
Does planet Earth turns slowly? It depends on how you look at it. At the equator it’s roaring along at more than 1000 miles per hour. This is pretty fast by terrestrial standards, and perhaps what Owl City had in mind. But on the other hand its angular velocity is just one rotation per day, or a very sedate 0.000694 RPM. Kids sometime imagine the earth as a spinning basketball and wonder why we don’t fly off, and that’s pretty much why: spin a basketball at 0.000694 RPM and I bet any ants on its surface will keep their grip just fine.
But this suggests a question: what would count as “fast” for the purposes of our scientifically inquisitive musicians? I think a good threshold might be the speed at which the earth’s rotation would start throwing people off at the equator. More specifically, the point at which the gravitational force required to hold people to the earth’s surface is exactly equal to the centripetal force required to hold them to the circular motion.
To be clear, there is no magical centrifugal force throwing people off, it’s simply that rotation is not straight line motion, and motion in anything other than a straight line requires forces. When standing on the ground there’s two of them. There’s gravity pointing down and the force of the floor (the “normal force”) pointing up. Together, they have the equal the exact centripetal force needed to keep you in circular motion about the center of the earth as the earth rotates.
We may remember that centripetal force is just the radius of the circle times the square of the angular velocity, and we may remember Newton’s equation for the gravitational force. If we do, we know they’re equal at the point where people start floating off their feet due to the rapid rotation of the earth. Why not write this relationship down?
Is it tough to solve this for the angular velocity? Nope!
Here M is the mass of the earth, m is the mass of an object on it (it’s canceled out in the last expression), G is the universal gravitational constant, and r is the radius of the earth. Plug in and you’ll get an angular velocity of 0.0012395 Hz. This isn’t too enlightening since it’s in radians per second, but if we turn it into revolutions per day it’s more interesting. Doing this (check my work please!) and I get a value of 17.045 revolutions per day.
So if the world rotated 17 times faster – in other words, a “day” of about 85 minutes – people near the equator would in fact go flying off. So would everything else, leaving us with a planet rapidly diffusing in space. Seventeen doesn’t sound like too much of a huge number, but then remember that centripetal force is proportional to the square of the velocity. Seventeen squared is a much more hefty 289 times more force. I think it’s fair to say planet earth would not in fact be turning slowly in that case, and Owl City’s worry would be justified. But as it is, I’d say it’s slow enough.