Math Fun

Your continued evasion merely proves my point.

OK, found the a problem:
[tab]A = C+G
B = H+D
B = I+F
C = G+E
D = H+J
E = G+I
F = J+D
H = I+J
A+B = C+E+F
A+C = B+H+I+F
A+C = B+H+J+F and/or
A+C = B+H+I+E
A+G = B+H+I

I’ll have to power through solving for something in terms of J later.[/tab]

Carleas???
[tab]10 Squares Puzzle Solution.png.png[/tab]

A fun paradox for you:

[youtube]http://www.youtube.com/watch?v=mW-0bZwoGwQ[/youtube]

Interesting, a little mind bending, and certainly difficult to express. Here is an attempt:

[tab]The circles are rotating at different speeds, and slower than the wheel. If there we more tick marks on the two circles and on the lines they are ‘riding’, it would be clear that the surface of the circle is moving slower than the string as they pass. That ‘slippage’ accounts for the difference in apparent circumference (the length of the string) and actual circumference.[/tab]

Yep. =D>

Now try to explain where the energy is coming from in this:
[youtube]http://www.youtube.com/watch?v=m8-Kek8Halc[/youtube]

And this:
[youtube]http://www.youtube.com/watch?v=_HV9YWmGf8g[/youtube]

First one
[tab]I think here the magnet acts as a fuel source, a source of stored energy and it will gradually lose effectiveness as the polarity of its atoms moves out alignment over time.
After writing that, I looked it up, and it is wrong… :blush:

:-k

OK, second guess: the potential energy from moving the ball away from the magnet is released as the ball moves towards the magnet.[/tab]

Second one
[tab]Here, I’m pretty sure the initial twist of the spring is what’s driving the spinning, and that that energy will gradually be lost as heat.[/tab]

I think that you missed on both counts:
[tab]1) The ball doesn’t move toward the magnet.
2) Note that for every turn of one end of the spring, the opposite end gets exactly one turn. The amount of twist in the spring must remain constant.[/tab]

[tab]In both, there is a subtle periodicity, i.e. they aren’t in smooth motion, but are slightly jerky. In the spring example, that means that the torsion moves through the system like a wave, so that one part twists a bit, then the next, then the next, so that the twist is actually travelling around the system in a loop. It seems the wooden handles act something like a diode, making it so that the twist can only ‘flow’ in one direction.

In the magnet/ball example, the ball rocks back and forth, drawn toward the magnet, and then moves backwards by the spinning wheel. The motion is started when the magnet is pulled away from the ball, so that the magnetic force decreases slightly and the ball falls, starting the wheel spinning. The ball then rolls, which spins the wheel. The wheel itself speeds up and slows down; as it spins faster, the ball is moved farther from the magnet and the magnetic force on the ball decreases, slowing the wheel and increasing the force.

So the ball is moving towards the magnet – and then away, and then towards, and then away…[/tab]

[tab]Ooo! I’ve been had! Treachery!! Fool me a half dozen times, shame on me #-o

Nice tricks though, clever setup. The one with the spring seems to have a motor in the frame, it looks like hits a switch with his right hand just as it starts spinning. I can’t figure out the other one, but I’m thinking it has something to do with the silver block used to hold the magnet.

The magnet car was a bridge too far, I knew that one was no good.

Thanks for making me providing me the opportunity to look like an asshat, James. =D>[/tab]

[tab]Nonono… you are selling yourself short. These are from VEProjects. They don’t use tricks. They merely don’t tell you everything so that you can figure it out. There are no hidden motors involved[/tab]

[youtube]http://www.youtube.com/watch?v=_HV9YWmGf8g[/youtube]
Spoiler:
[tab]The question proposed in the video asks if the torque at the servo spindle (right side) is greater than it is at the generator spindle (left side). The equation for the torque at the servo is:

t{g} = F * cos((π/2 – A) * sin(g)) * √¯((cos(g) * R / sin(A))² + (R * sin(g))²)

wherein;
g is the rotation angle of the generator spindle from top position,
F is the force applied by the generator spindle,
R is the length of the generator spindle radial arm, and
A is the angle of the servo spindle from horizontal upward.

With that, you should be able to answer half of the puzzle question: “Where is the energy coming from?
… although always check my math … it is just an estimate. :sunglasses:

Edited[/tab]

As it seems that no one is going to get this one:

The first spindle, A, has a torque advantage while it is positioned at the top and at the bottom of the spin circle. But at each side, the torque advantage becomes negative, a disadvantage. By integrating the proper formula for the torque advantage, one can see that the total advantage summed around the entire circle is zero = no advantage.

The question was, “from where does the energy come?” Obviously it has something to do with the twisted spring, but interestingly, the spring always has the same number of twists, implying that it isn’t giving up the energy within the twists. No matter how many times the spindles rotate, there will always be the same number of twists in the spring.

The answer for this one is actually similar to that for a pendulum swing, except with a pendulum swing the arm never actually return to the same position it started. But the pendulum begins to swing because the initial position is one of higher potential energy. The pendulum must start at the top of the swing. Similarly, the spindles must start at the top of the rotation circle where spindle A has temporary advantage.

Because of the temporary advantage, Spindle A forces a rotation to begin and the length of the spring prevents the motion of spindle B from directly countering spindle A. For a brief moment, the B end of the coil spring is more tightly wound than the A end. And by the time that greater tightness catches up to the A end, spindle A is no longer at the top where it had advantage but rather at the first side (3 o’clock) being slowed down by its new disadvantage. So the extra boost from the B end of the spring as a momentary pulse helps spindle A to get around the negative arc. The initial advantage of A gets passed through the spring back for A to use during its disadvantage arc period. And at that point end B is back to the same amount of twist with which it began.

During the lower portion of the rotation, the same thing is occurring. Spindle A has an advantage which it uses to put extra twist back into the B end of the spring which propagates around to help A out during its next upcoming negative arc (at 9 o’clock). Do the math.

So does it continue forever? No. There is friction using up the initial potential energy and eventually the spindles stop turning during one of the negative arc periods (3 or 9 o’clock) even though there is still the same number of twists in the spring as in the beginning.

So the answer as to where the energy came from is that it is “momentum pulse energy” provided by the system beginning at a high potential energy state, just like a pendulum swing starting from the raised pendulum position. And the number of twists in the spring is analogous to the gravity field of the pendulum.

[youtube]http://www.youtube.com/watch?v=nVSM__R1bqs[/youtube]
This is the video that convinced me that these videos are clever tricks and can’t be taken at face value.

Look in the upper left at the shadow changing, indicating some movement off camera. Then at 1:05, a wand of some kind comes into the top of the video above the spinning disk. That suggests that what’s really going on is that someone is moving a wand above the disk to generate the motion, perhaps using a focused stream of air, or a fine thread attached to the disk.

After this, their credibility is shot. Looking at their other videos, none would be impossible to fake, and many have what look like tells in light of the skepticism this video engenders.

Did you not understand my prior explanation?

This contraption is basically the same as the pendulum as well. He has to get the device in motion (the initial energy), in order for it to run. The mechanics are then balanced enough such as to allow that initial energy to keep flowing until the tiny bit of friction eventually uses it up.

It’s no big deal, no trick. You are looking for the ghost in the machine, like the American Indians fearing the magic spirit pushing the rail train. The magic isn’t there and they do not claim that it is. These are merely physical science puzzles made complicated by clever people trying to challenge perpetual motion … and getting close. It is just fun physics games.

There are a great many “almost perpetual” mechanisms that rely on either getting the device in motion, leaving it with momentum to drive it for a while, or starting the device at a high energy potential point in a cycle such that it will start off gaining momentum until another part of the cycle. In this video, the cycle doesn’t change from high potential to low or negative, so he has to push it in order to get it started.

Don’t let your paranoia trick you into seeing ghosts or magicians that aren’t there.

There is different i]category [/i]of mechanism that truly emulates perpetual motion because it absorbs energy from the surrounding air or fields. You are not seeing any of those kind from these people. But those are not breaking the conservation of energy laws either. They merely accentuate an inherent imbalance in potential energy. It is the “Second Law of Thermodynamics” that can be, and often is, broken. The “First Law of Thermodynamics”, the conservation law, is never truly broken although it can be made to appear to be, and effectively be, broken.

(Based on a true story)

Suppose you have an uninsulated hardwood floor that acts as a uniform heat sink for the room. You also have a blanket with an area (A) equal to the area of the floor. Will you keep the room warmer by laying the blanket flat across the whole floor, or folding it to double thickness and putting it across half the floor?

I have an intuition, but I don’t know the right answer; I’m more interested how people reason about this. Will post my thoughts when I have a minute tomorrow.

Did you mean for this to be a puzzle or did you just want the answer, or the explanation?
[tab]Treat the materials as resistors
Floor = R0
Blanket = R1

Total floor area = A

Blanket unfolded;
Floor resistance alone = R0/A
Blanket resistance alone = R1/A
In series,
Ra = (R0+R1)/A

With blanket folded;
Floor resistance bare = R0/(0.5A) == R3
Blanket doubled on floor = (R0+2R1)/(0.5A) == R4
Total bare floor and doubled blanket resistance;
[b]Rb = R3
R4 / (R3+R4)[/b]

For any positive value of R0 and R1, Rb will be lower resistance to heat flow, thus less insulated.[/tab]
But I suspect the more critical variable is what you are doing on that blanket. :sunglasses:

I did mean it to be a puzzler, because for me, with a poor knowledge of thermodynamics, I can only use my intuition and reason (assuming I am capable of using those :slight_smile: . So pardon the rambling I’m about to do, and please point out where I’m wrong (not that you’ve ever held back).

[tab]Your answer matches my intuition, that the blanket spread out would do a better job of insulating, but I don’t fully understand your math. Am I correct in reading you as calculating the total resistance of the floor? What is the equation you’re using in the folded blanket example? I’m just not sure how you’re calculating parallel resistance there.

And is that really the best way to calculate it? It doesn’t seem obvious that changing insulation on one part of the floor will affect the cooling effect of the other parts, but isn’t that the case for resistance, at least where the two halves are treated as parallel paths?

My intuition was to think of the problem in terms of rates, and ask how the blanket would affect the rate of heat flow. If an insulator decreases heat flow proportionally, i.e. by x%, then doubling the blanket would be less effective: The first layer of blanket would decrease the rate by x%, and then the second layer would decrease that lowered rate by x%. I suspect this is how insulation works.

If, however, an insulator reduced the rate by a fixed amount rather than a percent, then doubling up would be no different.

Finally, if insulation increased non-linearly, so that 2x the insulation produced >2x the reduction in rate, doubling could reduced total heat flow. The situation I’ve come up with where this might be the case is one where the blankets have a checkerboard pattern of open and closed spaces, so that when doubled they are fully closed. This could create a super-linear increase in insulation if a large obstruction to airflow is more effective than several small obstructions that sum to the same area.[/tab]

Your intuition is the best solution because a precise answer would have to account for radiative, convective, and conductive heat transfers requiring more knowledge of your blanket and the floor in terms of air leakage and thermal conductivity.

When insulating an area, your biggest bang will come with eliminating air leaks between insulating pockets of air, so even a sheet of saran wrap over the entire floor would do a better job than a blanket folded in half and it’s why people put plastic over their windows. Anytime you can divide one area from another area to prevent the volumes of air from intermingling, that’s half the insulation battle. The other half is slowing the conduction of heat from one air pocket to the other via the material used for the division. One way to accomplish that is to slow the turbulence of the air in the pockets which is the purpose of the fiberglass stuffing. If air cannot move around, it cannot convect heat and it slows conduction through the barrier considerably which allows a mere paper backing to be a good-enough barrier dividing the 2 volumes. (Plus glass happens to not pass IR light easily, which is insulative as well.)

Heat, as you think of it (and not the proper definition) is infrared light which is only different from visible light in that it oscillates a bit slower. All light comes from the vibration (acceleration) of charged particles. So keeping your vibrating charged particles inside your room is of utmost importance. As molecules vibrate, they are emitting IR radiation (ie cooling). When the IR meets another charge, it induces a vibration and continues on its way. The charge then emits its own IR in all directions resulting from the induced vibration and etc, etc, etc from particle to particle. Now if you let the particle float out of the room, you’ve lost all that energy. Also, if you let that particle touch another particle, it will transfer all its energy immediately to the other particle. The best strategy is to keep all your particles still and let them vibrate (cool) slowly over time. So reduce air turbulence and seal all leaks.

You could put a tarp down and then a blanket over it, but it would be much more effective to put the blanket down first and the tarp over it because you will have created a pocket of air between the floor and the tarp wherein the stuffing inside the blanket would reduce air turbulence and essentially mimic typical pink insulation. You see how it works?