Feynman's Infinite Pulley Problem

[Yuqing]

While looking for interesting problems, I was led to a problem involving infinite pulleys.

The problem and solution can be found http://www.feynmanlectures.info/solutions/infinite_pulleys_sol_1.pdf" . Now I follow the solution up to the part where he claims $\frac{1}{4^{i}M_i}$ is always vanishing (tends to 0) as i approaches infinity. This does not seem valid.

From $$M_{i-1} = \frac{4M_{i}t^{i-1}}{M_i + t^{i-1}}$$ we have
$$M_{i-1}=\frac{4t^{i-1}}{1 + \frac{t^{i-1}}{M_i}}$$

Since we know $$\frac{t^{i-1}}{M_i} \ge 0$$ we then have
$$M_{i-1} \le 4t^{i-1}$$ or equivalently
$$M_i \le 4t^i$$
from this we can get
$$\frac{1}{4^{i}M_i}\ge \frac{1}{4}\cdot\frac{1}{4^{i}t^{i}}$$
so that $\frac{1}{4^{i}M_i}$ is vanishing only if $\frac{1}{4^{i}t^{i}}$ is. I'm not sure how to determine the actual convergence of the M_is and I might be missing something but this solution does not seem to be valid because of the above point. Does anyone know of a way to perhaps salvage this solution?

 

[K^2]

Doesn't matter. If 1/(4t)ⁱ doesn't vanish, the problem diverges either way. So you can assume 1/4ⁱM_i vanishes without loss of generality.

 

[Yuqing]

My point is that $\frac{1}{4^{i} M_i}$ doesn't always tend to 0 as claimed. Is there any guarantee that $\frac{1}{4^{i} M_i}$ can't be non-zero even when $\frac{1}{4^{i} t^ {i}}$ is?

 

[K^2]

Yes. If you add blocks one at a time, 1/(4ⁱM_i) goes to zero whenever 1/(4t)ⁱ does.

 

[codelieb]

I would like to point out that this is not "Feynman's Infinite Pulley Problem." It's "Gottlieb's Infinite Pulley Problem!" invented many years after Feynman left this world. It just happens to be posted on http://www.feynmanlectures.info" . I also solved the problem (in fact, for arbitrary masses and any number of pulleys, making the posted problem only a special case), with some help from a friend, however I never published my solution. Sukumar Chandra, with whom I was corresponding at the time, and who liked the problem very much, was kind enough to write up a solution for me, which you can find posted with the problem.

Years later I discovered that David Morin had published a similar problem in his (wonderful) book, Introduction to classical mechanics: with problems and solutions, and he has also posted it online in his http://www.physics.harvard.edu/academics/undergrad/problems.html" . So I wrote to David about it:

Such (unphysical) "infinite machines" can have interesting and unintuitive properties -- in particular, with regard to the acceleration of m[0], which exhibits strange phase transitions dependent on the geometric factor 'a.' (When a = 1/2, all the pulleys are balanced and nothing moves, but when a < 1/4, mass m[0] free falls as if there were no weights at all on the other side of the pulley!) What is more interesting is that finite systems of pulleys and masses exhibit similar behavior, as I discovered by solving the general problem with N pulleys and arbitrary masses. Attached to this email is a letter I wrote to some friends stating that general solution. If you assume all the masses are equal, and take the limit of this solution as N goes to infinity, you will find that the acceleration of mass m[0] agrees exactly with your solution to the "Infinite Atwood's machine" problem, g/2. I furthermore demonstrate in the letter (using graphs generated with Mathematica) that systems of 10, 100, and 1000 pulleys with weights in geometric progression as described above exhibit similar phase transitions.

David replied:

Thanks for your email. Those strange behaviors are quite fascinating. I hadn't thought of doing the problem for an arbitrary ratio of masses. The flat behavior for a<1/4 is very interesting. I redid the problem with the "effective gravity" solution, and the a=1/4 special behavior does pop out. For a<1/4, it looks like there are two solutions. One is T=0 everywhere, then other has negative T. I guess this could technically be achieved by having stiff (against compression) strings, and also having them run through some sort of tubes on the top half of the pulleys so that they will remain in contact with the pulleys. But I need to think about this more. The transition at a=1/4 seems to be similar to the case where a block has friction with a surface, but if the block leaves the surface (assuming there can't be a negative normal force), then the behavior enters a new regime.

I have attached the letter referred to in my correspondence with David, in case anyone is interested.