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In the Feynman lectures vol 1, chapter 4, he mentioned a "Brilliant, clever" way of solving a problem named:

A simple example is a smooth inclined plane which is, happily, a three-four-five triangle. We hang a one-pound weight on the inclined plane with a pulley, and on the other side of the pulley, a weight $W$. We want to know how heavy $W$ must be to balance the one pound on the plane.

After solving the problem with a simple insight on conservation of energy, he describes:

enter image description here

It can be deduced in a way which is even more brilliant, discovered by Stevinus and inscribed on his tombstone. Figure 4–4 explains that it has to be $\frac{3}{5}$ of a pound, because the chain does not go around. It is evident that the lower part of the chain is balanced by itself, so that the pull of the five weights on one side must balance the pull of three weights on the other, or whatever the ratio of the legs. You see, by looking at this diagram, that WW must be $\frac{3}{5}$ of a pound. (If you get an epitaph like that on your gravestone, you are doing fine.)

After trying for around three days, I can't understand how you solve the problem with the Stevinus' epitaph (but I understand partially why the three weights on the vertical side should balance the five weights on the slanted, but they are same weight)

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  • $\begingroup$ It may be noted that I don't have any such epitaph on my gravestone and this video : youtube.com/watch?v=nDKGHGdXLEg helps a lot to understand partially. $\endgroup$
    – user77648
    Apr 1, 2016 at 0:46

5 Answers 5

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The ball-chain is intended to represent uniform weight distribution along to two upper sides of the triangle; if they were 10 times smaller you would have 30 on the vertical face, and 50 on the ramp. The hanging loop completes the chain.

Does the chain begin to rotate on its own? If it does, we have found a mechanical perpetual motion machine! So no, it doesn't move.

What about the forces at the two ends of the hanging loop? The tension must be the same, or the hanging loop would be pulled to one side -- and the chain would start to move.

Since the tension is the same on both sides, we can cut off the hanging loop, and the rest of the chain will remain stationary, for it was already balanced.

Also see History of Mechanics of the Inclined Plane, and the many footnotes.

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    $\begingroup$ So how does this argument solves the one pound - pulley problem ? $\endgroup$
    – user77648
    Apr 1, 2016 at 4:11
  • $\begingroup$ It's the same ratio, 3:5, as the lengths (number of balls); hence W=3/5. $\endgroup$ Apr 1, 2016 at 10:15
  • $\begingroup$ Can't resist: But... the masses at the top see a smaller gravitational force because they're farther from the Earth's center! :-) (yes, I know that's incorrect) $\endgroup$ Apr 1, 2016 at 12:16
  • $\begingroup$ @CarlWitthoft: Simon Stevin didn't know that; but it also makes no difference if the weights are uniformly distributed. :) $\endgroup$ Apr 1, 2016 at 12:28
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Looking at the diagram, you can see there are five "units of weight" along the diagonal, and three units along the vertical. We need to prove that these two sets of weight are in equilibrium; because, if they are, then "five units along the diagonal" is in equilibrium with "three units along the vertical", and thus if I have 1 unit along the diagonal (1/5th of the original), then I need 3/5 of a unit along the vertical.

So - why is this in equilibrium? Let's look at the chain as drawn. Should it move? If it would freely move clockwise, then I can put a very small generator on it to extract power - because as it moves, weights travel up one side as they come down the other, and this means that after it's moved one "click" (one unit of weight spacing) we have the same situation as before. Ditto if it moved anti-clockwise by itself. We conclude that the whole system is in equilibrium, or we have a perpetuum mobile. And we don't...

Would it still be in equilibrium without the bottom part? Well, the chain below the triangle is symmetrical. That means that the tension applied on the left must equal the tension on the right. And if these are the same, then removing the chain at the bottom shouldn't change the equilibrium.

And there you have it. This means that without the chain at the bottom, the thing is still in equilibrium: five along the diagonal equals three along the vertical.

Clever Stevinus.

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    $\begingroup$ I am very sorry, what you wrote I had understood relatively better (please see the video in the comment in the answer) than any other thing in the chapter. What I don't understand is the connection of the problem of the weights with this (and the first paragraph of your answer). $\endgroup$
    – user77648
    Apr 1, 2016 at 2:13
  • $\begingroup$ I wonder at what point in my answer I lose you? $\endgroup$
    – Floris
    Apr 1, 2016 at 2:20
  • $\begingroup$ Most part is relatively better decipherable except this portion: "if they are, then "five units along the diagonal" is in equilibrium with "three units along the vertical", and thus if I have 1 unit along the diagonal (1/5th of the original), then I need 3/5 of a unit along the vertical.". I am repeating, What I don't understand is the connection of the problem of the weights with this (and this portion of your answer). $\endgroup$
    – user77648
    Apr 1, 2016 at 3:11
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    $\begingroup$ The original problem asked about the ratio between two weights that would be in equilibrium. If there is a weight of one unit along the diagonal, what weight is needed along the vertical to balance it exactly. The weight "chain" gives us a ratio of 3:5 between vertical and diagonal. $\endgroup$
    – Floris
    Apr 1, 2016 at 10:33
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7 years late, and also a beginner in classical mechanics and physics in general, but I thought I'd give this a go.

First, what do we know?

Something that Feynman left out is that we define every bead in the chain to be equal weight. Therefore, if each bead is 1 lbs, the beads on the ramp add up to 5 lbs and the beads on the vertical add up to 3 lbs.

Next, we know that the force (tension force) acting down on both sides of the chain (the ramp on the right and the vertical slope on the left) is equal. This is easily derived from common sense since if one side of the chain were heavier, we would have a perpetual motion machine that would not only do work infinitely, but also create energy from nothing (a free energy generator, if you will), since there would be a constant net force down on one side of the chain.

So, what are we trying to find?

We are trying to find a weight of the beads representing W on the left vertical in order for the beads on the vertical and the ramp to be in equilibrium (no tension on either ends of the chain). Well, what a coincidence! The beads on the vertical and ramp have to be equilibrium or there would be a free energy generator. So, if each bead were 1/5 lbs, the three beads on the vertical representing W would be 3/5 lbs, and the five beads on the ramp would be 1 lbs.

And there you have it: an easy, simple way to find W without having to do any conservation of energy math or, even worse, having to individually calculate the tension in each bead.

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We can suppose another scenario where there is not incline, and the only way for the system to be balanced is for both sides to have 5 equal-mass spheres. But now there is an incline, so we know that the incline’s normal force is contributing to some of the lifting work to balance the system, thus, we don’t need 5 spheres to balance 5 spheres on the incline, we can have less to make the same effect. (Note that all along the spheres represent average weight distribution on the incline) Therefore it is easy to conclude W<M. But by how much? From the triangle we know that 5 spheres on the incline have the same effect (pull) as 3 spheres vertically hanging. Reversibly, we can say that you only need 3 spheres to pull on 5 spheres on the incline. Therefore, 3/5 of the pull on the incline. Its like only using 3/5 of the amount of work done by the other side to achieve the same effect.

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because it is a 345 triangle. If the triangle were rotated and the beads were on the 4 and 5 sides it would be the same, 4/5 of a pound, answer. If it were a flat surface the answer would be 0/5, meaning the beads would go nowhere. Make sense?

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