bongostaple » October 6th, 2016, 2:49 pm wrote:
At the point where the straight path ball has traveled to the Pi marker, the circular path ball is just a little past the three-quarter point of the circular path.
Bearing in mind that the two sections of tube in question were cut at the same length, the only reasonable explanation, as far as I can see, is that the circular path ball has slowed down.
If I were to take the position proposed by Mathis/Videobloke, wouldn't it be equivalent to saying that the tube may well be pi*d long when it's straight, but if you bend it to a circular shape then it is now 4*d long?
In a sense, yes, it is equivalent to that, even if that is not an accurate description of what is going on. But that is why it is so hard to understand or wrap your head around. It's as if the tube is getting longer just by bending it into a circle. How is that possible? But you see it's not that the length of the tube changes. It's that in order to move around the circle, the ball has to travel a longer distance. When he says in his post that 'you'll feel like you're in a netherworld,' I think it's precisely this apparent 'warping' of space-time that he's referring to. To be clear: there is no warping, but the explanation for what we're seeing is fairly deep and described in depth in his various papers on pi, Newton's lemmae and the virial theorem.
As for the two sections of tube being cut to the same length: they weren't. Unfortunately, the video doesn't show the circular tube laid out straight, but he does measure the diameter as 17.6cm, so the length of the tube is 17.6cm * 3.14 = 55.264cm. The length of the straight tube from the beginning of the circle (0) is 70.4cm.
70.4/55.264 = 1.273885.
4/3.14 = 1.273885
So the ratio of the length of the tubes are the same proportion as 4 to 3.14.
What this means is that, as the straight ball travels 17.6cm (1 diameter in length -- the tick marks), the ball in the circle only travels 1/4 of the circle, which we can measure as having a length of (55.264/4=) 13.816 cm. But what Miles is trying to argue is that even though the measured length of the tube is only 3.14, the distance it travels (length/time) is 4. Another way to say this is that the ball in the circle only moves 3.14 in the time it takes the ball in the straight to travel 4, even though they are moving at the same speed.
That would leave two possible explanations for the results as they are presented:
a) The ball loses speed when it's forced into a circular path.
Or
b) A piece of plastic tubing gets longer if you shape it into a circle.
I really can't see how b) could be more likely than a) as an explanation, but I'm happy to consider both angles.
As I tried to show above, the option is C: the ball travels a larger
distance in a circular path even though the tube remains the same
length. Again, the explanation for this and the distinction he draws between length and distance can be found in his papers. I encourage you to take the time to read them.
I appreciate your note of centripetal force, but in the circular situation, it is balanced with the normal force offered by the wall of the tube. This in theory should result in the ball and the tube wall, for want of a better phrase, 'pressing together harder', and I think this would increase the friction involved.
The friction of gravity holding either ball down in the direction of the tabletop can be regarded as equal, but the circular path ball has some extra friction, and I believe that must slow it down during its travel.
So in the scenario you are sketching, the ball is being pushed up against the outer wall of the tube (by virtue of centrifugal force), then a centripetal force acting on the ball would push it away from the outer wall of the tube. Since the centripetal and centrifugal forces are 'equal and opposite', the ball should not be pressing harder against the tube. In any event, if there were an increase of friction, we would see the ball getting progressively slower. I have watched the video, and I don't see evidence of this. I watched it again just now, and while I do think the circular ball hits the halfway mark just a smidge after the straight ball hits the 2 diameter mark, they both seem to hit their marks at the same time by the third quarter / 3rd diameter marks. So no apparent slowdown in my view. The mark at the halfway might just be a little bit off. Again, more precision in the experiment would be very welcome.
I thought of a situation where there is a four-wheeled cart on a level track and you give it a push to a known speed, it travels a distance and friction eventually brings it to a halt.
Now let's say we add some extra weight to the cart by getting Miles Mathis to climb aboard. We then give it a push until it's going the same speed as when we let go of the unburdened cart.
Will the cart travel the same distance with Mathis on board?
I *think* the extra weight = more friction, but on the other hand, assuming it's up to the same speed, the cart+Mathis combo will have more momentum.
I'm continuing to ponder on this one - it just doesn't feel right that the circular path ball could have its course changed through a full 360 degrees without losing some energy somewhere. Nobody coasts around a Wall of Death - they are constantly putting more energy into the system to achieve the same constant speed - so why would this be different?
I think whether the cart will travel the same distance with added weight depends on the mechanics of the setup. If the cart is on wheels on a relatively smooth surface, then I think the added momentum would carry the day. But if the cart is, say, not on wheels but just sliding on a surface, then the added weight might add significantly to the friction so he would go a shorter distance. In short, I think it would depend on the specifics of the situation.
As for constantly putting more energy in the system of achieve a constant speed -- well we do that moving in a straight as well as "around a Wall of Death." You cannot coast forever in a straight line, either. So both of the balls we see are slowing down. Don't forget also that in the Wall of Death you also have gravity pushing you down. The question is whether the curved and straight paths slow down at the same rate. Fortunately this is an empirical question: just add a straight end to see if the ball exiting the circle is traveling the same speed as the straight ball. If friction had slowed it down, then it will not magically regain speed moving into the straight path; it would remain at the slower speed.
My first thought when I saw your picture is "I need to get my eyes checked."
I don't agree with your conclusion. It might hold for 1 and 3 if 1 starts the curve immediately at the point and ends the curve immediately at the other point. So in other words, if the curve is a 1/4 circle. From the way you've drawn it, it's not clear. The other condition would be that 1 is able to change direction without a loss of speed. So travelling both paths at a constant speed.
But 2 doesn't work because it's partly straight path and partly curved. If we take his running track as an analogy, lane 1 spends more time in the curve, so the distance (not length!) is greater than 2, which spends more time in the straight. So 1 and 3 should both take longer than 2.