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Gravitational Experiment

11/24/2006 3:26 PM

Hello all--Here is a thought experiment that crossed my mind the other day and I don't have enough education to know how to obtain the correct answer. I am a new user of CR4, so if this is an innapropriate use, please inform me.

Anyway, imagine an ant in a small container attached to the tread of an automobile tire. The vehicle to which the tire is attached is going down the road at some constant speed. This is a perfect tire that does not deform at the road service--it does not heat up or expand or change dimensions in any way. The ant in his little apartment is at the edge of the rubber and does not get squashed or flung away or any of that other real-world stuff. We can even ignore earth's gravity constant since it will be the same I think, (unless we go to velocities approaching C) for both of the obvious ways of looking at this experiment.

Here's the question: What acceleration forces (indescernible from gravity to the ant)does the ant experience?

From the point of view of an observer in the vehicle, he should feel a constant centripetal thrust sensed as being away from the center of rotation of the tire.

From the point of view of a roadside observer, he should feel a vertical accelration upward through a distance equal to the radius of the tire and then decelating to a momentary stop, again through a distance equal to the tire radius, before accelerating and decelating downward in the same fashion, to the beginning point, over and over according to the sine function. He should simultaneously feel similar forces in the horizontal direction accelerating and decelerating between stopped and twice the velocity of the vehicle.

Witty responses are enjoyed although I really do appreciate thoughtful analysis from which I may derive new insights.

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#1

Re: Gravitational Thought Experiment

11/24/2006 4:27 PM

I am confused.

If the car or tire is rotating at a constant velocity, the only acceleration vector is the centripetal one and it will be constant, despite the fact that the car is rolling forward.

For an observer in the car they will only feel the acceleration produced gravity because he and the car are moving at a constant velocity.

For an observer outside the car standing by the road he will only feel the acceleration produced by gravity and a slight breeze as the car goes by (please note the author has attempted to interject a small amount of wit into this analysis. It is left as an exercise to the reader to discover where this wit has been planted).

For the ant there is no gain and or loss of acceleration as he haplessly spins around the tire, only a constant force induced by centripetal acceleration. The ant may experience dizziness, although, but I do not know the physiology of an ant. Perhaps there is an entomologist in the group that could expand on this.

If you include the effect of gravity from the Earth for the ant, then the net acceleration felt by the ant would be the sum of the two vector accelerations. One would be the radial acceleration (centripetal); the other would be the force of Earth's gravity. If you graph the resultant vector for one revolution of the wheel, the predominant vector would be centripetal modulated slightly by Gravity.

Does that make sense?

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#2
In reply to #1

Re: Gravitational Thought Experiment

11/24/2006 7:20 PM

You wrote: "I am confused." ...stuff... "Does that make sense?"

For a confused person, your post makes perfect sense (except for the wit part). So, what sorts of things do you post when you're lucid?

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#3
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Re: Gravitational Thought Experiment

11/24/2006 7:40 PM

When and if that ever happens, sir, I shall let you know!

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#4
In reply to #1

Re: Gravitational Thought Experiment

11/24/2006 11:37 PM

The effect uv gravity woudnt be slight. There woud be a 2G difference between the top and bottom uv hiz rotation. He woud feel a continuous rotary oscillation at the rate the tire wuz rotating.

To the original poster, the ant woud feel no effect uv the forward motion uv the car.

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#5

Re: Gravitational Thought Experiment

11/25/2006 4:47 AM

If I understand you correctly, lshurtle, your question has to do with different reference frames, three of them: the ant's, the passenger's and the bystander's.

My take on it is (ignoring gravity, as you said): in the ant's frame, he is stationary and observes the car spinning around him and feels a constant acceleration in one direction - as you said: "indiscernible from gravity".

In the passenger's frame, the ant is spinning and suffers a constant centrifugal force in a changing direction.

In the bystander's frame, the ant suffers a constantly changing acceleration in a constantly changing direction and will in fact be stationary at the lowest point and moving at twice the car's speed at the top. To put this into an equation is not simple, but doable!

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#6

Re: Gravitational Thought Experiment

11/25/2006 5:28 AM

If you ignore Earth's gravity, then the ant's body weight will be perpetually pressed in to the tire all the way around and the ant's feet will resist the centrifigal force of it's body weight by pressing againsed the tire. No other forces come into play. However, the ant's hart beat will actually slow from General Relativity effects but, in reality, will probably speed up from concern over its situation.

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#7

Re: Gravitational Thought Experiment

11/25/2006 6:33 AM

What you have here is a classic example of adding two vectors. The first is constant and is gravity while the second is rotating and is the acceleration caused by the rotation of the tire see below.

What we have is the force of gravity (shown in red) acting down and the centripetal acceleration caused by the rotation of the tire (shown in blue). The centripetal force can be broken up into two constantly varying forces, one horizontal (shown in green) and the other a vertical one (shown in purple). Now the horizontal component will not be effected by gravity and will oscillate forwards and backwards (green sin wave). The vertical component however need to be added to gravity and depending on the speed of rotation may or may not completely overcome gravity. In this example the rotational forces (purple sin wave) don't overcome gravity and as a result the waveform crosses the 0g line.

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#28
In reply to #7

Re: Gravitational Thought Experiment

11/28/2006 7:18 AM

Hi Masu: I'm curious. What program did you use to make the nice sinusoidal waveforms and graphics? Looks pretty neat. JEJ

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#29
In reply to #28

Re: Gravitational Thought Experiment

11/28/2006 1:17 PM

It's a really expensive drafting package called DeltaCad. You can download it over the net and a license is about $40. It takes a couple of hours to learn how to drive it but it's only a drawing package so its only as good as your drafting skills.

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#30
In reply to #29

Re: Gravitational Thought Experiment

11/28/2006 6:08 PM

Thanks: I'll check it out when I have a chance. JEJ

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#8

Re: Gravitational Thought Experiment

11/25/2006 7:21 AM

Instead of an ant on a tyre, imagine you are in a car in a continuous loop-the-loop fairground ride.

You would feel 'light' during downward motion and 'heavy' during upward motion.

Where at the top of the loop, the car must be going fast enough to overcome gravity not to drop off the track.

Which would happen if you follow the purple line in MASU's example.

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#9
In reply to #8

Re: Gravitational Thought Experiment

11/25/2006 7:49 AM

Horace40 wrote: "You would feel 'light' during downward motion and 'heavy' during upward motion.

Where at the top of the loop, the car must be going fast enough to overcome gravity not to drop off the track."

Guys, you are all doing excellent analysis, but are you not answering the wrong question? lshurtle said explicitly: "We can even ignore earth's gravity constant since it will be the same I think, (unless we go to velocities approaching C) for both of the obvious ways of looking at this experiment."

In doing your thing, you lost sight of the real question, I think, which I tried to answer in post #5. Shoot me if I'm wrong! lshurtle please help!

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#10
In reply to #9

Re: Gravitational Thought Experiment

11/25/2006 8:28 AM

The original post says;

"We can even ignore earth's gravity constant since it will be the same I think, (unless we go to velocities approaching C) for both of the obvious ways of looking at this experiment."

which is a tad confusing. I took this to mean that we can assume gravity remains constant rather than ignore gravity altogether, which as Jorrie has pointed out is the literal interpretation.. Either way all the answers have been given so maybe Ishurtle can select the appropriate response and clear up the confusion.

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#16
In reply to #9

Re: Gravitational Thought Experiment

11/25/2006 5:27 PM

Exactly, Jorrie!

"Here's the question: What acceleration forces (indescernible from gravity to the ant)does the ant experience?"

The answer is: the ant only feels centripetal force.

It doesn't matter where the observer is, who the observer is, or what velocity the observer is traveling at. The ant only experiences centripetal force.

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#18
In reply to #9

Re: Gravitational Thought Experiment

11/26/2006 6:49 AM

JORRIE said. Guys, you are all doing excellent analysis, but are you not answering the wrong question? lshurtle said explicitly: "We can even ignore earth's gravity constant since it will be the same I think, (unless we go to velocities approaching C) for both of the obvious ways of looking at this experiment."

In doing your thing, you lost sight of the real question, I think, which I tried to answer in post #5. Shoot me if I'm wrong! lshurtle please help!

Sorry about that. I now appreciate the gravity of my error.

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#22
In reply to #9

Re: Gravitational Thought Experiment

11/27/2006 1:38 AM

But isn't the real question as follows?

Here's the question: What acceleration forces (indescernible from gravity to the ant)does the ant experience?

The fact that the original poster says "I think" after asserting that we can ignore gravity suggests that perhaps we can't. Certainly, there are not a lot of physical phenomena happening close to earth's surface in which gravity can be ignored. Although it is difficult to know exactly what an ant can and cannot discern, I think it is reasonable to assume that, for the purposes of this question, it could discern the difference between a constant force and an oscillating one which could cause its weight to halve and double periodically.

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#27
In reply to #22

Re: Gravitational Thought Experiment

11/27/2006 10:27 PM

Thanks for all the input--When I first formulated the question, my thought was that if the ant knew the earths gravitational constant, could he subtract or cancel that acceleration from the others that were acting on his little world and theoretically derive the existence of the tire, the car, etc. Obviously, a corollary to myself in a sealed capsule (basement bedroom perhaps?) with adequate sensitive accelerometeric instrumentation (bet I spelled that incorrectly--oh well) being able to derive the existence and motion of the earth, sun, galaxy, etc.

I am stunned by the intellectual horsepower being brought to bear on my poor little ant's situation. How very, very neat!

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#11

Re: Gravitational Experiment

11/25/2006 9:30 AM

The curve described by a point (ant) attached to a tire rolling along the ground is called a CYCLOID and has the following parametric equations: x=a(Θ-sinΘ), and y=a(1-cosΘ). Where a is the radius of the wheel.

These equations describe the path traced by the point on the circumference as the wheel rolls along the ground. To find the velocity of the point along its path, you have to take the first derivative of these equations and to find the acceleration you have to find the second derivative. In order to find the total acceleration experienced by the point (ant) you have to add or subtract the acceleration of gravity at each point along the path traced by the point.

An interesting sidelight about Cycloids is that Galileo first called attention to the Cycloid, once recommending that it be used for the arches of bridges. Pascal once spent 8 days attempting to solve many of the problems of cycloids, such as finding the area under one arch, and the volume of the solid of revolution formed by revolving the curve about a line. The cycloid has so many interesting properties and has caused so many quarrels among mathematicians that it has been called "the Helen of geometry" and "the apple of discord".

I hope this throws some light on the subject.

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#12

Re: Gravitational Experiment

11/25/2006 1:26 PM

What wonderful responses. I am indebted to you each one. Actually, I realized that I had posted a relatively trivial (read "dumb") question about ten minutes after I had posted but I didn't know how to get it back. Jorrie, your analysis comes closest to what I was considering although I hadn't any concern about the passenger in the vehicle. The sine wave graphing and pictorial analysis offered by...Masu?..I think...yes, was right on target. And I very much appreciated the historical info on cycloids from bhb. Also thanks to the rest of you for your "lucid" exploration of the idea. And some humor doesn't hurt a bit.

Actually, my thought was to determine if the ant ("point") had any chance of perceiving the different acceleration forces acting upon him--with obvious correlation to our situation on the face of a revolving planet orbiting a sun while whirling around a galaxy as it speeds through intergalactic space in a straight line (whatever that means) as we go about our business traveling here and there in planes and trains, etc.

He doesn't.

Vector analysis and summation of the acceleration forces in the horizontal and vertical as observed by a roadside observer exactly equals those derived from analyzing the constant radial acceleration experienced by the ant on the tire. I don't suppose this gets interesting until you assign a velocity of 1/2 C to the vehicle so that the top of the tire approaches light velocity and the ant's mass approaches infinite (along that axis at least)--everything probably cancels again--I don't have the math to go there.

Thanks all for participating

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#19
In reply to #12

Re: Gravitational Experiment

11/26/2006 4:04 PM

Other than Hero, there seems to be a general assumption that the ant could not tell that it was rotating. What if we have a vehicle with small wheels that is speeding along an autobahn, and a container fixed to the axle?

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#20
In reply to #19

Re: Gravitational Experiment

11/26/2006 6:11 PM

If you're a physicist you ought to know that the ant is not rotating, it is jumping up and down like a grasshopper on steroids, except a grasshopper's trajectory would be a parabola (in a vacuum) and the ant's trajectory is a cycloid. It does not see the rotation of the wheel just the arc going up and down as the wheel rolls along the ground, coming to a complete stop every time the point where the ant is located touches the ground.

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#21
In reply to #20

Re: Gravitational Experiment

11/26/2006 6:46 PM

I thought I put a question mark after the moniker - but it didn't come through, and I appear to be stuck with an arrogant moniker. So I'd be grateful if someone could tell me how to change that.
This ant has been moved up (in its frame of reference) to the axle, so it doesn't stop. Not that that stopping makes any difference to the ant.

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#24
In reply to #21

Re: Gravitational Experiment

11/27/2006 4:20 AM

Hmm. Even up here on the axle, the poor insect is still experiencing g-forces that pin it to the 'floor' of its container. Perhaps we should slow down a bit. That's better, 2g, and the ant is beginning to move. I'm glad I installed a time-stretching camera system. It's walking in the direction opposite to the spin of the wheel, and it's starting to climb the 'side'. Whoops it's fallen off - but I'm sure that wall was not even vertical. Now its inspecting itself - front legs, middle - whoops it's rolling along the floor. Now it's turning sideways - oh no it's not, it's rearing up. Do ants always behave like this, and I've never noticed? Will someone tell me what is going on?

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#25
In reply to #24

Re: Gravitational Experiment

11/27/2006 5:19 AM

I forgot to say - I was watching the ant so carefully that I drove off the edge of a cliff (I had to leave the autobahn). All the activity described above happened shortly after the car started to fall. All of which is a long-winded way of saying that any explanation can ignore the earth's gravity.

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#36
In reply to #19

Re: Gravitational Experiment

04/01/2008 1:58 PM

There are no rotational forces at the centre of rotation.

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#37
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Re: Gravitational Experiment

04/01/2008 2:31 PM

Read on - you'll find that my point (sic) was that an ant has finite dimensions.

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#38
In reply to #37

Re: Gravitational Experiment

04/02/2008 10:02 AM

Boy, talk about a time warp. With the last post prior to the recent blurb being some 16 months ago I thought this thread was well and truly dead and buried.

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#39
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Re: Gravitational Experiment

04/02/2008 10:07 AM

Aren't X-aliens supposed to be capable of raising the dead? and perhaps these things come around..

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#23
In reply to #12

Re: Gravitational Experiment

11/27/2006 1:55 AM

You say: He doesn't.

I say: He does.

Most analyzers seem to take on faith your assertion that "We can even ignore earth's gravity constant since it will be the same I think… " (italics mine). However, we can not ignore gravity if we are interested in knowing the nature of the forces acting on the ant. Clearly, at very low wheel speeds, gravity will predominate.

Newton could tell us the directions and magnitudes of the forces acting on the ant, given a road speed and a tire diameter. However, if the problem is couched in seemingly relativistic terms, a good deal of confusion enters in. For example, to suggest that the acceleration forces experienced by the ant are "indiscernible from gravity to the ant" (presumably because the ant is believed to be in its own little world, with its own inertial reference frame) is simply wrong, because the ant will experience rapid cyclical accelerations most unlike the relatively constant forces of gravity. At low road speeds, there will come a point in the tire's rotation where centripetal force and gravity cancel, and the ant would feel weightless at that point, and heavier than normal at another. At high road speeds, centripetal force will always exceed gravitational force, so the ant would feel, let's say, very heavy for part of the revolution and extremely heavy for another part. (I am using the term centripetal force loosely here, given that the ant's path will be a cycloid.)

To say that the forces on the ant are "indiscernible from gravity to the ant" is analogous to saying that the forces on a human on a Ferris wheel are indiscernible from gravity to the human. The human can discern that he/she feels lighter over the top and heavier through the bottom arc. A pilot flying a loop feels the same variation in forces, but more dramatically.

The point of view of an observer has nothing to do with the forces acting on an object. In other words, whether you view the ant's motion from inside the car, from alongside the car or from an airplane flying by, the forces on the ant are unaffected. Whether you watch the ant from in the car or outside the car has nothing to do with the ant. If you have some imagination and the ability to think in three dimensions, your observation point should have no affect on your ability to describe the forces acting on the ant. An articulate observer inside the car will describe the ant's trajectory (if he is concerned about the forces acting on the ant) in the same way as an articulate observer outside the car. The observer in the car might see the ant's path as a straight line (if the observer were directly behind the tire (and assuming the car has no body work of any sort) or an ellipse; but he could not see the path as circular, due to the difficulty of putting one's head in line with the wheel's axle while remaining in the car. So the observer in the car would have to think about and imagine the ant's movement through whatever reference frame makes sense for determining the forces in question. Given that gravity must be taken into account, then the path relative to the earth's surface seems most relevant.

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#13

Re: Gravitational Experiment

11/25/2006 1:55 PM

With regard to your comments on sine waves and centrifugal forces acting on the ant. If the car were standing still (jacked up) with the tires rotating, there would be a centrifugal force on the ant. However, since it is moving down the road, the motion is not sinusoidal and the only forces acting on the ant are the force of gravity and the acceleration caused by its motion in a cycloidal form. Let's say the movement starts from the lowest point (i.e. contact with the road) the movement is straight up starting slowly and increasing in velocity until it reaches the top of the tire after a half revolution. Now it is moving horizontally forward at twice the speed of the car, then it starts moving down till it again contacts the road. At this point the motion is zero, both horizontally and vertically. The only forces (aside from gravity) are those generated by the cycloidal motion, not centrifugal. If you want to calculate the forces, you have to take the second derivatives of the equations that give the "x' and "y" coordinates as mentioned previously. Does that make sense?

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#14
In reply to #13

Re: Gravitational Experiment

11/25/2006 3:08 PM

Yes, it does make sense, and thank you. Approaching it through differential calculus is in fact the way to precisely describe the perceived force at any given instant. I think what was of most interest to me is the fact the the result of the vector analysis of the cycloidal motion (what the bystander sees) is exactly equivalent to the result of the vector analysis of the circular motion (what the ant experiences)--ergo--the ant can't tell if he is traveling along the road or not. The car could be sliding sideways on ice and he would not be able to sense it--as long as the motion is constant (I think). Now I have to think about that ant when some acceleration is applied to the vehicle.

thanks for you come-back.

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#17
In reply to #13

Re: Gravitational Experiment

11/25/2006 5:42 PM

Well, the motion of the car is really unimportant because the car is in a constant state of motion where the vector of the car is unchanging.

The only vector that is changing is the vector that is rotating around the hub. By definition, that is acceleration (centripetal). It is dependent on the angular velocity.

If the car was slowing down or speeding up, then that would be a whole new ball game. That would then be the sum of the changing vectors

So, a vector simply represents velocity and direction. Acceleration is a change in a vector; either the vector's direction or a change in its speed or both.

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#15

Re: Gravitational Experiment

11/25/2006 3:35 PM

Where can I buy the tires?

I think it is an interesting question and I'll venture a quickly thought through opinion:since the vehicle is moving forward at a constant velocity there will be no lateral acceleration. The forces on the ant will be only the centrepital forces exerted by the outer section of his little compartment -the product of its own mass times the angular acceleration of the tire rotation[(2pi*frequency of rotation)^2 *Tire Diameter] plus and minus the force of gravity. So at a velocity and tire rotational speed is such that the angular acceleration divided by the gravitational acceleration = 1 the ant would be weightless(and float) at the top zero degree and weigh twice as much as normal at the bottom 180 degree zero point. They would look like a cycloid vs. position of the vehicle. Let's see what some of the professors think. JEJ

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#31
In reply to #15

Re: Gravitational Experiment

11/28/2006 6:55 PM

The positional motion will be (one of the) cycloids. But the acceleration is indistinguishable from the same object moving in a circle. The acceleration of a point on/in the tyre is as described by Masu. Because the ant (presumably) rotates witht the tyre, the acceleration/pseudo-gravity perceived will be a constant (due to the centipetal acceleration) plus a rotating vector (due to gravity) - as described by several repondents. The only things missing are that the centripetal component will vary in amplitude and direction at different positions on the ant's body. Plus, that the effect on the ant will change if it starts to move. In particular, if it pulls the ice-skater's trick (by starting in an extended position and then compacting itself into a ball) it's rotational velocity will increase. Neither the variation nor the change in rotational speed is likely to be noticeable if the ant is all the way out on the circumference of the tyre - hence my earlier suggestion of moving the ant's container nearer the centre of rotation. There are a couple of other effects (due to the ant moving) that I described that I haven't explained. (The effects as described are somewhat of an exageration if the ant is only at the edge of the axle, however - I think the ant would need to be in the order of twice its body-length from the actual axis of rotation to observe the full effect).

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#32
In reply to #31

Re: Gravitational Experiment

11/29/2006 1:17 AM

It seems reasonable to say that, as the ant tries to turn to the left, he might find himself pushed over onto his right side by gyroscopic effects.

Seems like it could be a wild ride for the ant. But perhaps the ant might worry that it must be very difficult for us to navigate without the ability to walk on ceilings.

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#33
In reply to #32

Re: Gravitational Experiment

11/29/2006 12:01 PM

We have adjusted the original experiment somewhat by moving the ant near the center of rotation, etc., but that does lead to some interesting observations. Originally, by removing earth's gravity from the statements regarding cycloidal and circular motion to arrive at the conclusion that the ant could not sense any difference, I essentially removed the car, tire, and ant from near proximity to the earth's surface as was pointed out by an early response. I then decided I must think about the car under a condition of acceleration. Well, "Duh", if the car is on the face of the earth and subject to earth's gravitational force, it is in fact under the influence of an acceleration and you were all way ahead of me. Because of that situation, there might well be forces acting on the ant in his little compartment that he could detect. So....

In reply to the excellent observations of Blink and Physicist and Guest then, perhaps if the ant had a pendulum hanging from his "ceiling" he would be able detect rotational and precessional effects. If he had an atomic clock he might be able to detect variations in the period of the pendulum (keeping all this in a strictly Newtonian universe for the time being), and if he had computer charting available, he might be able to develop a graphic depiction of the pendelum's length-of-arc and end points and velocity changes over a few million iterations.

It is interesting to me to speculate that such a depiction might look quite a bit like chaos theory "strange attractor" mapping and contain within it the information that, once deciphered, would give him a picture of the outside universe--all from his little windowless apartment.

What think you?

Lon

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Join Date: Oct 2006
Posts: 488
#34
In reply to #33

Re: Gravitational Experiment

11/29/2006 12:35 PM

Thank you - that is pretty illuminating. If you can keep the losses low enough, this should give you all the information about rotation that you could reasonably (or otherwise) hope. I just wasn't thinking in terms of such a sophisticated insect.
Regarding "chaos", I think the situation should be much simpler than for the standard Foucault pendulum: the period of the earth's rotation is vastly greater than the period of the wheel (order of 100k times as long*). It therefore makes sense to ignore the earth's rotation, and we initially need to consider just two of the times involved in the process - the period of the pendulum (short) and that of the wheel (sub-second).
*The equivalent ratio for the Moon's effective period to the Earth's (which characterises Foucault's pendulum) is about 28:1.
BTW, in the unlikely situation that anyone was worrying about the safety of such a negligent driver, it turned out that the "cliff" was a drop of only about a foot - just far enough to observe the ant-ics.

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Anonymous Poster
#26

Re: Gravitational Experiment

11/27/2006 7:23 PM

The ant believes he experiences a centrifugal force pulling him towards the outside of his apartment/edge of rotational orbit. In reality, he is acelerating and the apartment is imparting centripetal forces into the ant keeping him orbiting rather than flying off tangentially to the point of acceleration on the tire. He also feels the effect of gravity. The vector sum of the apparent centrifugal effect and gravity would equate to the net effect on the ant. If rotation was high enough gravity would represent a small increase and decrease sinusiodallly in the apparent weight of the ant applied to the apartment would increase and decrease accordingly. If rotation was small, the gravity would prtewntial overwhelm the centrifugal effect and the ant would feel he was falling up/down as his apartment appeared to be rotating, like clothes in a dryer.

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Associate

Join Date: Mar 2008
Posts: 42
#35

Re: Gravitational Experiment

04/01/2008 1:56 PM

Hi,

Let me see if I got this right.

There is this ant enjoying a ride perched somewhere on the circumference of a tyre (tire) of a car in motion and the question is what forces will Mr. Ant feel.

Now Mr. Ant, describe your trajectory.

Mr Ant replies, OK alien, I have just taken pencil to paper and established a new branch of mathematics I call the Calculus of Variations.

But Mr. Ant, Newton did that in one night back in the I don't know when. Never mind, what did you find out.

Well, says Mr. Ant, I found that my trajectory is a portion of a regular cycloid.

Is that a circle?

No, you stupid alien, it's nothing to do with circles and the forces I feel are nothing to do with centriwahtsits, petal or fugal.

Isn't fugal to do with Bach?

Just let me get on with my ride on this vehicle because obviously you are a dumkopf.

Regards,

alan.

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