Weightlessness at the Equator (Whiteboard Sketch #1)

The mathematics blog Scientific Finger Food has an interesting entry, “Weightlessness at the Equator (Whiteboard Sketch #1)”, which looks at the sort of question that’s easy to imagine when you’re young: since gravity pulls you to the center of the earth, and the earth’s spinning pushes you away (unless we’re speaking precisely, but you know what that means), so, how fast would the planet have to spin so that a person on the equator wouldn’t feel any weight?

It’s a straightforward problem, one a high school student ought to be able to do. Sebastian Templ works the problem out, though, including the all-important diagram that shows the important part, which is what calculation to do.

In reality, the answer doesn’t much matter since a planet spinning nearly fast enough to allow for weightlessness at the equator would be spinning so fast it couldn’t hold itself together, and a more advanced version of this problem could make use of that: given some measure of how strongly rock will hold itself together, what’s the fastest that the planet can spin before it falls apart? And a yet more advanced course might work out how other phenomena, such as tides or the precession of the poles might work. Eventually, one might go on to compose highly-regarded works of hard science fiction, if you’re willing to start from the questions easy to imagine when you’re young.

scientific finger food

At the present time, our Earth does a full rotation every 24 hours, which results in day and night. Just like on a carrousel, its inhabitants (and, by the way, all the other stuff on and of the planet) are pushed “outwards” due to the centrifugal force. So we permanently feel an “upwards” pulling force thanks to the Earth’s rotation. However, the centrifugal force is much weaker than the centri petal force, which is directed towards the core of the planet and usually called “gravitation”. If this wasn’t the case, we would have serious problems holding our bodies down to the ground. (The ground, too, would have troubles holding itself “to the ground”.)

Especially on the equator, the centrifugal and the gravitational force are antagonistic forces: the one points “downwards” while the other points “upwards”.

How fast would the Earth have to spin in order to cause weightlessness at the…

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The Box Drops

So the last piece I need for figuring out whether it’s easier to tip a box over by pushing on the middle of an edge or along one corner is to know the amount of torque applied by pushing with, presumably, the same force in both locations. Well, that’s almost the last bit. I also need to know how the torque and the moment of inertia connect together to say how fast an angular acceleration I can give the box.

A Second Way To Fall Over

I admit not being perfectly satisfied with my answer, about whether a box is easier to tip over by pushing on the middle of one of its top edges or by pushing on its corner, just by looking at it from the energy both approaches need to raise the box’s center of mass above the pivot. It’s straightforward enough, but I don’t do this sort of calculation often, so maybe I’m looking at the wrong things. Can I find another, independent, line of argument? If I can, does that get to the same answer? If it does, good. If it doesn’t, then I get to wonder which line of argument I believe in more. So here’s one.

How Two Trapezoids Make This Simpler

[ More of Trapezoid Week! Here we make finding the area simpler by doubling the number of trapezoids on the screen. ]

Figuring out the area of a trapezoid based on making it the difference between two triangles works all right. “All right” carries with it a sense of inadequacy. The complaints against it are pretty basic. The first is that it doesn’t work for everything which might be called a trapezoid. Maybe we don’t want to consider parallelograms and rectangles to be particular kinds of trapezoids, but, why rule them out if we don’t have to? The second point is the proof is a little convoluted, requiring us to break out of thinking about trapezoids to remember details of similar triangles. It’d be nice if we had a more direct way of proving things.