re: energy producing experiments

[Delburt Phend]

A perfect stop (for the 10 to one mass ratio cylinder and spheres) appears to occur at a tether length of  about .834 diameters of the cylinder. The backward motion of the cylinder is gone; it now stops and precedes forward. It is still four frames to cross the black square at beginning and end.

The cylinder stop is also dead center of the complete experiment. The tether at 90° to tangent occurs about half way between the two four frames (start and finish) velocities.

[Delburt Phend]

Oops; I forgot to add the radius of the sphere (center of mass), which is 12.75 mm. That would make the tether length .978 diameters instead of .834.

[Delburt Phend]

In one of the runs of the 10 to 1 cylinder and spheres it took four frames to cross the black square at the start and four frames to cross at the finish of the experiment. In the middle is a dead stop of the rotation of the cylinder.
 
Energy conservation predicts that at the end of the experiment it will take 12.6 frames to cross the black square. Because for energy conservation; when the spheres have all the motion, the spheres would have to be moving at the 'square root of ten' instead of ten times faster than the starting motion of the cylinder and spheres. Four frames time 3.16 is 12.6 frames.
 
Well I was curious: I found the middle frame, of the videoed experiment, where the cylinder is stopped. There are actually about five frames where the cylinder does not appear to move; but I picked the center of these five frames. From this middle frame I clicked off 12 more frames. By the end of the 12th frame one black square had been crossed from side to side. So, from the stop, the average velocity is at least 1 square per 12 frames. That makes the final velocity at least 2 squares per these 12 frames; final velocity is roughly double average velocity if you start from a stop.  This is twice as fast as predicted for energy conservation; and this is starting from a stop. These 12 frames start at the slowest portion of the experiment for the cylinder.

In the next twelve frames (that would be 13 -24 frames from the stop) there were 2.5 black squares crossed in 12 frames.  Now the average speed is 2.5 times faster than the max expected for energy conservation.
 
In a linear acceleration the final speed would be double the average speed of 2.5 squares; for 5 black squares per 12 frames (1.67 per 4 frames). The graph of this acceleration is not linear because it troughs out at 4 frames per crossing and there are several frames that have almost zero motion. The graph of this acceleration would probably be more like a section of a sine curve.

The point is that 12.6 frames to cross is way too slow to be the correct answer; and energy conservation is eliminated as a possibility. The direct measurement of four frames for momentum conservation fits perfectly.

This troughing out, or little velocity change for several frames on the graph, is logically expected. When the cylinder is stopped in this experiment and the tether is at 90° to tangent little (or no) rotational force can be applied by the tether.  And at the end of the experiment the spheres and cylinder are moving at the same speed; the spheres on the tangent tether are moving at the same speed as the rotation of the cylinder. Both these arrangement cause acceleration to cease.

[Delburt Phend]

I built 2 more cylinder and spheres machines; I converted the 10 units of total mass to one unit of sphere mass model. I made a 20 to one and a 30 to one by adding 1320 grams and then 1508  additional grams. The 1508 was stainless steel rods so they were left too massive.

I did a careful mass to diameter evaluation of the three cylinder and spheres models. The spheres are assumed to be point masses of 132 grams at a diameter of 114.3 mm; compared to the cylinder diameter of 88.9 mm; and so on for all the other mass. The greater diameter gives you greater speed; and four sets of mass are at different diameters.

So the 10 to one is actually a 9.78 to one.

And the 20 to one is actually a 19.33 to one.

And the 30 to one is actually a 32.87 to one.

I wanted to compare the tether length that causes a perfect stop of the cylinder at full extension; when the tether is at 90° to tangent.  I used the tether length that actually touches the cylinder.

I got 72.4 mm of tether for the 9.78 to one.

I got 146.4 mm of tether for the 19.33 to one.

I got 239.7 mm of tether for the 32.87 to one.

There appears to be a one to one relationship between the length of the tether and the mass the tether is able to stop.
 
I went back to the 4.5 to one total mass to sphere mass, cylinder and spheres, and shortened the length of the tether for a perfect stop; with this one to one relationship in mind. The 4.5 to one is actually 4.55: after the above diameter evaluation. So 72.4 mm tether length times 4.55/9.78 mass ratios would give you about 34 mm tether length.

I found that a tether length of 32.5 mm, for the portion of the tether that actually touches the cylinder, had a perfect stop at 90° to tangent.  This remains within the 5% error ranger.

This seems to confirm the one to one relationship between the tether length and the cylinder mass stopped; but lets put tether length in radial length of the cylinder.

A tether length of .731 radii stops 4.55 to one.   4.55/.366 = 6.22

A tether length of 1.629 radii stops 9.78 to one. 9.78/ 1.629 = 6.00

A tether length of 3.294 radii stops 19.33 to one.  19.33/3.294 = 5.87

A tether length of 5.39 radii stops 32.87 to one.   32.87/5.39 = 6.1

So if you want to have a perfect stop at 90° to tangent: for a cylinder that has a mass of 23 times that of the spheres you would use a tether length of about 4 radii. This is for any size cylinder. And the tether length is from the cylinder surface.

All lengths can restart the cylinder to the full initial speed.

[Delburt Phend]

Or is the relationship 2 * pi? And why?