Dear Straight Dope:
A couple of friends and I have run across a problem we have been unable to solve and it bugs me. Since you're the world's smartest human I'd appreciate you giving it a shot.
Given: Planet shaped like a donut.
Question: Would a man be able to stand on the inside rim or would he be pulled to the center of the donut? Or does it depend on the dimensions of the donut and if so, what are the ratios that determine whether he stays on land or floats towards the center?
Assumptions: No rotation of the planet. The only force is that of gravity.
Maybe you've heard this one, maybe you haven't, but we'd all be truly impressed if you knew or could find the answer. The best I've been able to do is approximate the forces involved and it's too close to call.
SDStaff Chronos replies:
We couldn’t get Cecil interested in this, but don’t worry, it’s not too hard. Remember from physics class, how you can treat a body like all of its mass is concentrated at the center of mass? OK, first of all, forget that. That’s only true when the bodies involved are spherical. To work with any other shape, you’ve got to break it down into pieces which are either spherical, or are small enough that their shape doesn’t matter. Then, for each piece, you use Newton’s equation, F = Gm1m2/r2 to get the force due to that piece (m1 is the mass of the piece, and m2 is the mass of the person or whatever). You then add up all of those forces, and you’ve got the total force.
What’s that, you say? That doesn’t sound too easy to you? Well, there are a few shortcuts you can use. First, as with any physics problem, you need to look at the symmetry of the problem. If you have a man on the inner surface of the donut, the force on him is either going to be pointing straight in towards the center of mass of the planet, or straight out towards the ground he’s standing on. Any force that would pull him to the left, for instance, is matched by a force pulling him to the right.
The second shortcut comes from looking at the equation, and seeing that the force is proportional to one over the distance squared. This really isn’t too surprising, since this is the way that most things spread out in three dimensions, including light. The upshot of this is that, assuming the donut has uniform density, you can determine the direction of the gravitational force just by looking around you. Wherever there’s the most mass in your field of view (assuming that you could see through objects), that’s where you’ll be pulled. When you’re standing on the inner rim of Planet Donut, almost the entire lower half of your field of view is filled by the ground, but the other side of the donut only takes up a small portion of the “sky,” so the ground you’re standing on is pulling you a lot harder. Your weight will still be somewhat less than if you were on the outer edge of the donut (since there is some gravity from the far side), but you won’t float away.
Just for kicks, let’s look at another science fiction scenario in the same way: Suppose you had a planet that was only a hollow shell of a sphere. What would gravity be like inside? Now, whenever you look around, you see the inside of the planet, no matter where you look. All the forces cancel out, so gravity is zero — you float. Hollow planets are pretty rare, but there is a practical application of this: if you’re in a cave deep under a planet’s surface, at distance r from the planet’s center. You can think of the whole planet as a solid sphere inside radius r plus a shell outside radius r. Since the shell has no effect on you, the gravitational force you feel is due solely to the portion of the planet below you. If the cave is at the center of the planet, gravity exerts no net force on you at all.
SDStaff Chronos, Straight Dope Science Advisory Board
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