# Feynman's Infinite Pulley Problem

1. Aug 28, 2011

### 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" [Broken]. 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 Mis 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?

Last edited by a moderator: May 5, 2017
2. Aug 29, 2011

### K^2

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

3. Aug 29, 2011

### 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?

4. Aug 29, 2011

### K^2

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

5. Aug 30, 2011

### 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" [Broken]. 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" [Broken]. So I wrote to David about it:
David replied:
I have attached the letter referred to in my correspondence with David, in case anyone is interested.

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Last edited by a moderator: May 5, 2017