Please cast your mind back through the mists of time to when I posted my blog Three puzzles to ponder. Someone just emailed me with a question regarding this blog to which I’m sure I know the answer, but…
As you may recall, the third puzzle was concerned with Gravity, which – of course – explains the title of this blog. What do you mean “What does gravity have to do with dogs and the groins of strangers?” What do they teach the young people of today? Who amongst us could forge the immortal words of the great American philosopher Dave Barry, who famously said: “Magnetism is one of the six fundamental forces of the universe, with the other five being gravity, Duct tape, whining, remote control, and the force that pulls dogs toward the groins of strangers.”
But we digress… the problem I originally posed was to assume that the circumference of the world is exactly 24,000 miles at the equator. We know that the earth spins round once every 24 hours, so at the equator this is 24,000 miles in 24 hours, which equates to a rotational speed at the surface of 1,000 miles per hour (no wonder my hair always looks so mussed up).
Anyway, the question I posed was as follows: "If you are standing at the equator, the spin of the world is sort of trying to throw you off (if you see what I mean). So, assuming that someone has a mass of exactly 100 kilograms if we were to weigh him or her at the North or South Pole, what would the reading on the weighing scale be if our subject was standing on it at the equator?”
Over time we added all sorts of qualifiers, such as the weighing scale being calibrated in such a way that being moved to the equator did not affect it, and so on and so forth.
The thing is that someone who shall remain nameless to protect the innocent – let’s call him Simon (you get 10 extra points if you correctly identify the animated film that prompted me to use the name Simon) – emailed me with a query, that spawned a flurry of messages, which evolved into a telephone conversation, whose conclusion left both of us scratching our heads…
Simon: There is of course the difference between mass (an inherent property of matter, independent of gravitational field, velocity, or acceleration) and weight (which is the force felt by the Earth pushing up on your feet, dependent on your mass and the Earth's gravitational field and any vertical acceleration). The mass doesn't change. The gravitational field of the earth doesn't change. Since there is no vertical acceleration, the measured weight will be the same. That's my theory, anyway.
Max: But the Earth is spinning, so the guy will weigh less at the equator than at one of the poles because the spinning earth causes centrifugal (or centripetal?) force...
Simon: Suppose you were standing at the equator when the speed of the Earth’s rotation suddenly doubled – you would fall over backwards because the acceleration would be in the horizontal direction – this is elementary stuff they teach at high school.
Max: I must have been off school that day. So here’s another example, suppose I have a rock on the end of a piece of string and I’m spinning it round and round really fast in a vertical plane such that it passes my feet and then goes above my head…
Max: So now let’s assume that the string breaks (or is cut by a razor) at the point when the rock is directly above my head. In this case the rock has a mix of inertia in the vertical direction and motion in the horizontal direction, so it will fly up and across, sort of thing…
Simon: Of course it won’t. At the point that the rock is directly overhead, the only motion it has is horizontal, so it will take off in a horizontal direction:
Max: You are making my head hurt. Let’s return to my original problem, but let’s assume that there is no atmosphere. Suppose I’m standing on the equator of an airless object like the moon that is spinning really, really quickly. Are you saying that I won’t weigh less at the equator than I do at one of the poles?
Simon: That’s right.
Max: Suppose that it’s spinning incredibly quickly – like 100,000 times an hour, surely it would fling me off into space.
Simon: No it wouldn’t because all you would have would be horizontal motion, but no vertical acceleration.
Max: Arrrgggh! Look, imagine that you’re looking down on the equivalent of the North Pole on the moon and you see me standing in a spacesuit on the equator. Let’s assume that, from your point of view, I’m standing on top of the moon, which is spinning at some rate in a clockwise direction:
Max: If the moon suddenly disappeared at this point in time, I agree that I would carry on traveling in a horizontal direction off to the right. Now, even if the moon doesn’t disappear, inertia will cause my body to want to continue travelling in a horizontal direction. However the gravity of the moon will pull me down, but due to my horizontal motion I will appear to be lighter – that is, although I will have the same mass I will appear to weigh less. If I gradually kept on increasing the rotational speed of the moon, there would come a time when my tendency to keep travelling horizontally (my inertia coupled with my horizontal velocity) would be more than the gravity of the moon and my feet would leave the ground.
Simon: No they wouldn’t.
Max: Yes they would.
Simon: No they wouldn’t.
Max: Yes they would.
… This is pretty much where we left things … with me obviously winning the argument (grin) …
Actually, now I come to think about it, I could use that Physics modelling program Phun to model the one about the string with the rock (Click Here
to see my blog on Phun). Well, I could if I had the time, but as usual I am up to my ears in alligators fighting fires without a paddle ... maybe you could use Phun to test this out and report back to the rest of us...