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Centrifugal Force And Human Hamster Wheels


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This is a rather simple problem to someone used to Mechanics Calculations--at least so I would imagine.

 

Lets say that I wanted to build a Man-Sized Hamster Wheel in my back yard, to exercise in. Lets make it a little wider than it absolutely has to be, for comfort sake. The larger Diameter that I can make the wheel, the more my running track will approximate running on flat land, or up a gentle slope.....

 

But if we make it too large, both cost and construction problems will multiply.

 

Assume that I'm an average runner--in rather good Road Running shape.

 

Is it possible for me to run fast enough to measurably increase the apparent gravity I'm running against?

 

Not for a few fleeting instants, but for at least twenty minutes to a half hour?

 

And by Measurable--less than 5% would be pointless, but I'm looking for at least 10%.

 

Is there any possibility that I might run my little wheel fast enough to cause any circulatory or stroke problems?

 

I mean, I doubt it--but the head would be riding in a much tighter orbit and experiencing less Centrifugal Force than my feet.

 

So what Dimensions would my wheel have to have, and how fast would I need to run to get the Exercise Benefits of running in a higher "G" Field?

 

Not really wanting to build it--but I've been imagining it for years---and I'd like to be able to put some Real World Numbers into the Daydream.

 

Saxon Violence

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Is it possible for me to run fast enough to measurably increase the apparent gravity I'm running against?
Ordinarily, you don't run against gravity unless you run up a gradient. So, what exactly do you mean by 10% more? It isn't trivial to compute the effort of running on flat ground.

 

That said, it depends on how much resistance the wheel offers to being turned at a rate corresponding to a reasonable running pace. If its bearings are too free it only offers resistance while increasing velocity. Whatever resistance there is, it implies you need to run at a spot shifted from the lowest point and this amounts to running uphill. The force can be computed according to body weight and the lengths involved.

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If you’re thinking of a simple scaling up an actual hamster/mouse exercise wheel into something like this

(from http://www.jasonlane.org.uk)

you won’t get any increased downward force on your body, because your body isn’t in circular motion. The wheel is – someone strapped to it would feel both the usual force of gravity and a centrifugal force – but the person running in place on it wouldn’t.

 

If you want to experience increased force due to your own running motion, you’d need some variation of a banked circular track, like this

post-1347-0-39922000-1332965020_thumb.jpg

, with as small a radius as possible allowing you to fit inside it.

 

Let’s throw some numbers at this to get a rough approximation:

A person can run about 5 m/s

Let’s try a 2 m radius

Centripetal acceleration would then [imath]\frac{5^2}{2} = 12.5 \,\text{m/s/s}[/imath], about 1.3 g, horizontal. Vector adding with the vertical 1 g pseudo-acceleration from gravity, we get [imath]\sqrt{1.3^2 +1^2} \dot= 1.6 \,\text{g}[/imath].

 

If your height were only a small fraction of the track’s 2 m radius, we could call the approximation done now, and say you’d have to lean inward about [imath]90 - \arctan \frac{1}{1.3} \dot= 52 ^\circ[/imath] (so the track should be banked something close to that), but this isn’t the case. So let’s assume your mass is concentrated at your center of gravity, say 0.9 m from the bottom of your feet. We could try to write an exact equation now, but that’s too hard for me, so I’ll just iterate.

 

Reduces the radius about [imath]\cos (90 ^\circ -52 ^\circ ) \dot = 0.71 m[/imath], to about 1.29 m, and the speed by the same ratio, to about 3.2 m/s. Centripetal acceleration is now [imath]\frac{3.2^2}{1.29} = 8.5 \,\text{m/s/s}[/imath], about 0.87 g, our combined pseudo-acceleration, [imath]\sqrt{1.3^2 +1^2} \dot= 1.32 \,\text{g}[/imath], and our lean angle about 39°. Repeat this adjustment a few times (easy with a decent calculator), and it settles down to a radius of 1.4 m, speed 3.5 m/s, net pseudo-acceleration of 1.34 g, and a lean of 42°.

 

We can iterate a bit more by reducing the track’s radius, but don’t gain much: 1.41 g at 45° for a 1.3 m radius track, where the top of the head of a 1.8 m tall runner would be nearly stationary.

 

Now, whether anyone but a talented, trained acrobat in her or his prime could actually run on a track like this without constantly staggering, falling, running off it, etc, and how nauseating it would be (not only are you lapping the track about every 2.5 sec, your head, middle, feet etc. are experiencing different force vectors in both magnitude and directions – pretty much a perfect, staggering, puke summoning storm, I imagine), I can only guess, but if you could, you’d be road training at 1.3 to 1.4 g.

 

The problem here is that we humans are too slow to have much centripetal acceleration running around a track with a radius many time as great as our height. To get a usable high-g training room, you’d need to simply spin a big room around a very big radius circle via a very long arm or track – essentially a scaled-up version of the centrifuges used to test/train some pilots and astronauts, like this one:

 

I imagine such a thing would break anybody’s build budget. ;)

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