Weightlessness at the Equator (Whiteboard Sketch #1)

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 centripetal 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 equator?

This question can be put another way just as well: At which rotational speed are the absolute values of the centrifugal and the centripetal force the same?

Calculations show for which value of the circulation period T the stick person becomes weightless.

Calculations show for which value of the circulation period T the stick person becomes weightless.

The answer to the question is: If the Earth rotated about 17 times faster than it currently does, equator inhabitants would be weightless indeed. A day-night-cycle would last only 1.4 hours instead of the familiar 24 hours – in other words: 1 hour and 24 minutes. The equatorial speed around the Earth’s center would increase from normal 1668 km/h to almost 28500 km/h (from 1036 mph to 17700 mph). Phew!

Astronauts aboard the International Space Station (ISS) experience this kind of day-night-rhythm every day (or rather every …umm…seventeenth-day) and see a sun rise every one and a half hours. And of course, they are weightless in their orbit.
(It’s quite reasonable to apply our conditions for weightlessness on earth surface to the ISS since its orbital height of approximately 400 kilometers is numerically small compared to the earth radius of 6400 kilometers – thus, the uncertainty in our statements is in the region of just a few minutes.)

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About tempse

I think about physics, other stuff, and physics. Besides, I share some thoughts on the internet.

Posted on March 8, 2014, in Fun Fact, mechanics, physics and tagged , , , , . Bookmark the permalink. 2 Comments.

  1. Reblogged this on nebusresearch and commented:

    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.

  2. michaeldavidsutherland

    Thank you, this helped me.

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