I don't know if you saw my last post directed to you in the other thread or not. It was covered up pretty quickly by other stuff, but anyway, in that post I addressed the Smokin' Lamas and analyzed a bit of data from a youtube video of one of the pumpkin tosses.
In that particular one it appears the launch speed was about 176 ft/sec or close to 54 m/sec when the wheel was spinning at around 360 rpm. I compared the interval traveled between frames with the wheel diameter which I assumed to be about 6 feet, so the calculations are only approximate. Maybe you have more information on the actual dimensions?
Anyway, I'll just go ahead and quote that post here so others can at least see from my gif animations the device you are speaking of.
http://www.besslerwheel.com/forum/viewt ... 039#118039Furcurequs wrote:pequaide,
Before I get back to addressing jim_mich's mistaken math, which he still seems to be in denial about to some degree, I thought I would just go ahead and address your stuff since I already had most of this post written.
In the thread I spoke of in which jim_mich had that mistaken math, you seem to have also been mistaken in yours. We know why yours is wrong, though, as I and apparently several others have tried to explain to you before. You leave one of "r"'s out of the moment of inertia equations - while apparently just choosing not to accept what are accepted fundamentals of physics.
Now, with that said, to address one of your questions about the Smokin' Lamas:
We know that the distance of 1776 feet is the horizontal distance that the pumpkin traveled while it was in the air. During the travel time, though, it also both ascended and then descended in the vertical.
So, to get a rough ball park figure for the minimum launch speed, we can easily do some calculations assuming no air resistance and that the pumpkin was launched from the same vertical level as that on which it landed. For maximum range with no air resistance we would have also wanted to have launched at a 45 degree angle, too.
That means that the magnitude of the launch velocity (v) multiplied by the sine of 45 degrees will give us the magnitude of the velocity in the vertical and the magnitude of the launch velocity (v) multiplied by the cosine of 45 degrees will give us the magnitude of the velocity in the horizontal.
We know that the velocity in the vertical is slowed to a stop during the ascent by the acceleration due to gravity (g) and then the acceleration due to gravity (g) then speeds it back up in the opposite direction. Since "g" is a constant, we know that the time of ascent must also equal the time of descent (again, assuming no air resistance ) and that the final speed in the vertical will be the same as the initial speed but in the opposite direction.
Let's now set up our equations:
v * sin(45degrees) - 32.2 feet/second^2 * t = 0 feet/second
In the above equation "v" is the magnitude of the launch velocity and "t" is the time of ascent - or the time it takes the pumpkin to come to a stop in the vertical direction.
We now have one equation and two unknowns, so we need another equation:
v * cos(45 degrees) * (2 * t) = 1776 ft
In this second equation "v" is again the magnitude of our launch velocity and "t" is again the time of ascent, so the speed in the horizontal multiplied by the total time in the air equals our horizontal distance traveled and so I'm multiplying "t" by 2 in this equation since the time of ascent and descent are equal.
Now we can solve for the initial velocity "v" (and leaving out some gory details):
v = ( ( 1776 feet * 32.2 feet / second^2 ) / (2 * sin ( 45 degrees) * cos ( 45 degrees ) ) )^0.5
v = 239 feet/second or 163 miles/hour
t = 5.25 seconds , so total time in the air is about 10.5 seconds
Anyway, that should be a pretty good ball park figure for a minimum launch speed. The actual launch speed would, of course, have to have been greater due to air resistance losses. Also, one might want to consider and possibly correct for differences in the vertical height of the launch and the landing positions if one had more data.
I actually went seeking some data and found this nice video of a side view of a Smokin' Lamas pumpkin toss:
http://www.youtube.com/watch?v=CsjzJKKhMlU
I extracted the videos frames of just the launch and put them in this slow motion animated gif:
You can see that the pumpkin is quite clear and that there seems to have been very little motion blur. So, I decided I'd just overlay the pumpkin from several different frames onto one and got this:
While the pumpkin is in the air the camera also seems to be pretty stable, too, so that was quite fortunate.
Here is the same thing but with the sling from the previous three frames also overlaid, but unfortunately I've noticed that it appears there may actually be a missing - as in a "dropped" - frame from the video right around the actual time of release. There is more than twice the distance between the first view of the pumpkin in the air and the last view of the pumpkin in the sling than between any other two shots of the pumpkin in consecutive frames, so unless at the release the pumpkin lost over half its speed and over 3/4ths its kinetic energy - which almost certainly wouldn't have happened - there is indeed a problem with the video.
Anyway, if we just look at the intervals between pumpkins in the air, though, we can see that the first apparently reliable interval is very close to the same distance as the diameter of the wheel. Guessing at around a 6 foot diameter for the wheel (after having seen pictures of people standing next to it) and using the frame rate of 29.33 frames per second, that would put the speed of the pumpkin at about 6 feet * 29.33/second or approximately 176 feet per second which works out to be about 120 miles/hour.
In this animated gif I made from frames taken from immediately before the launch you can see that the wheel turns just slightly more than a full turn in 5 video frames.
Here is an animated gif using only the zeroth and fifth frame from the above and so if you look at the end of the red spiral you can see just how close to the same position they are. I chose these frames so you can use the spot of sunlight and/or the line there in the background as a reference.
Anyway, this would indicate that the speed of the wheel is around 360 rpm.
It's hard for me to find specific details, but I did find on the Smokin' Lamas Facebook page that the latest wheel (which I'm not really sure that this is, btw) weighs in at 665 pounds.
If the wheel in this video is the 665 pound version and the mass were distributed uniformly as if in a solid cylinder, then at this rotational speed the wheel should have had over 10 times the energy that the flying pumpkin ends up with. If the mass were mostly in the rim, it would have been over 20 times the energy of the pumpkin.
So, most of the wheel mass may actually be concentrated near the center here.
You said the tether length doesn't matter, but by drawing a line through the pumpkins shown in the air and then drawing a line through the center of the wheel that is at right angles to that, we can determine the moment arm and calculate the angular momentum of the pumpkin around the center of the wheel.
I wish I had more details about the wheel itself, like the actual mass distribution and where the pumpkin rides in the wheel before it is launched, so that I could properly model the whole thing, but one might be able to get approximations for those things using equations based upon the conservation of energy and the conservation of angular momentum.
After thinking about it, if these folks knew their physics they could probably reduce the wheel mass significantly and get the same results. With a (2x) bicycle wheel type design with most of the mass near the rims, they could maybe reduce the weight to nearly 1/10th of what they have. It would be a lot easier to transport and mount, I would think.
Of course, though, without really knowing specifics about what they really do have, I'm just sort of speculating. There may also be some strength considerations that come in to play, too.
Oh, also, from reading the rules of the main punkin chunkin contest that they enter, it seems the person pedaling their wheel up to speed only has a maximum of two minutes in which to do that.
From what I've read, weightlifters can for very brief periods put out nearly 3 horsepower in an anaerobic effort. More sustained aerobic efforts like that of cyclists, though, max out at only a little over half a horsepower.
Highly trained cyclists can supposedly output around 400 watts for as long as an hour. "Highly trained," though, in cycling probably also includes the taking of illegal performance enhancing drugs and blood doping, if you keep up with such things. ...haha
Also, with actual cycling the body is cooled from the motion through the air which ups the sustainable power output too, since the body itself is essentially a heat engine that works more efficiently when kept cool.
Anyway, if the guy doing the pedaling was putting out about 1/3 horsepower for 2 minutes to bring the wheel up to speed, that would be about 3 times the energy the pumpkin has here - if my quick calculations were correct.
In other words, they are not getting something for nothing.
Also (and as even with Bessler's wheel) it would be rather nice to have some data on the air resistance of an actual spinning disk like that to know how much energy was wasted just getting the thing up to speed.
Pequaide, I do like your experiments and I think it would probably be a good thing to better understand how quickly and efficiently energy from a flywheel can be transferred to a projectile using a tether like that, but still, there seems to be no sign of any excess energy in these experiments and they are quite well modeled and explained using traditional concepts and math. ...if, of course, one has all the correct parameters.
BTW, I just put out over 1/2 a horsepower myself doing A chin-up! I'd need to do 120 in two minutes, though, to be able to sustain that. Maybe I should move to jumping jacks. ...or squat thrusts?
Take care.
Dwayne
Dwayne