Understanding Feynman's argument about Stevinus

Question

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:

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)

Answer 1

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.

Answer 2

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.

Answer 3

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?