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...
Put a ping pong ball on the edge of a turntable that plays LPs (if you have one). Set it up so that as the turntable turns the ball stays in the same place. Then give the ball a little poke to push it into the center. Watch it go in a circle. Figure out why and I think you'll have your answer.
@corndodger: "There is no "horizontal" force..."
Yes, the word "horizontal" was misused -- in each case, the intended word was "tangential".
@corndodger: "Cut the string and the ball flies away from you in the direction it was pointing when the string was cut."
If you're going to be pedantic about it, the ball will fly away in a direction perpendicular to the "direction it was pointing when the string was cut." The ball continues in the direction it was moving, not pointing, at the instant the centripetal force was removed.
@corndodger: "Sit in the center of a spinning merry-go-round, ball in hand. Let go of the ball. The ball leaves, rolling straight out from the center where you're sitting..."
Actually, barring friction with the surface of the merry-go-round, in your rotating reference frame, the ball will appear to slide outwards in a *trailing spiral* since it's velocity remains constant but it's angular velocity decreases as the radius increases. Add friction, and the ball's outward spiral gets deformed due to the rotational inertia of the ball as it comes up to speed with the surface of the merry-go-round.
As for your word "centrifical", the word for the stipulated force allowing you to treat a rotating frame as an inertial frame is "centrifugal".
Perhaps you should consult something a little more advanced than _Sandbox Science_ before you go correcting others.
I was taken aback by you discussion of a "horizontal" component to the ball being swung around your head and feet. The ball must have hit your head a couple of times too many.
There is no "horizontal" force, only the centrifical force that pulls the ball away from you, stretching the string. Cut the string and the ball flies away from you in the direction it was pointing when the string was cut. It does not follow the circular path it was in prior to release. This information can be found in any pre-school physics book. (This is too basic for Kindergarten students.)
Want further confirmation? Sit in the center of a spinning merry-go-round, ball in hand. Let go of the ball. The ball leaves, rolling straight out from the center where you're sitting, and continues rolling straight away once it is on the ground.
Air pressure, more specifically, air density and thus buoyancy, would have an effect, but it would be immeasurably small. The rate at which you lose mass due to sweating would likely have a greater effect.
I once posed a puzzle along the lines of "Which weighs more at STP: a kilogram of gold or a kilogram of aluminum"; I was looking for the buoyancy effect, but found that the oxidation of the outer surface of the aluminum increased its weight by more than the decrease due to buoyancy.
@ck_02: Well, it's just so crazy it may work.
Did you read my Discworld blog (http://bit.ly/mRLJN5). Your comment reminded me of the scene in Guards! Guards! Where our three heroes are on the roof with one about to shoot an arrow at a dragon to hit its "vulnerable spot" and one says something like "it's a million to one chance but it just might work" and another says "it has to work because a million to one chance always works in stories."
But then one remembers that the guy who is going to fire the arrow won a competition in archery and is also going to use his lucky arrow .. and after some debate they decide that it's more of a thousand to one shot, and whoever heard on that working in a story?
So they start to handicap the archer by having him stand on one leg and close one eye and ... until they recon they've managed to get the odds back up to a million to one ...
I love this stuff :-)
However, if a ventilation system were in place that kept constant pressure and air volume into account while providing said subject with a measured amount of breathable air that would have to be taken at the time of each weigh-in. Well, it's just so crazy it may work.
Relative to any difference that might occur and be measurable in the effective weight due to your position on the globe, the donut factor, as alluded to by ck_02, would be an order or two of magnitude greater in impact.
Certainly, sealing yourself into a plastic bag would prevent the donut factor from influencing your measurements. However, the plastic bag would in a short span of time, lead to your inability to continue with your measurements. Unfortunately, it would not make you a closed system as plastic does, at very small rates, allow the permeation of moisture and various gasses.
In conclusion, I would not recommend sealing yourself into a plastic bag. While it would prevent some influences, it would ultimately fail in creating a completely closed system and therefore would call your data into question.