## Bessler's Belt.

Johann Bessler (1680 – 1745), a.k.a. Orffyreus, compiled a book Machinen Tractate containing pictures of an astounding variety of perpetual motion devices known at his time. Among them we find this small picture (printed upside down in the book) of a belt-style device. The belt runs over fixed pulleys (A and B) and has articulated arms with weights on their ends, hinged at the belt so that on one side they hang freely close to the belt, but on the other side they are held extended at right angles to the belt. Therefore the net gravitational torque on them is greater on the left than on the right.

The net weight on the left side (C) is equal to that on the right (D). But what about the torques? The weights on the left have greater lever arms than those on the right so the torques seem to be unbalanced. Then why doesn't the wheel turn counter-clockwise continually?

The short answer: If the belt is manually turned, he work gained by weights descending on the left equals the work required to lift them on the right. No net gain. It won't initiate or sustain motion.

Also, Stevin's Principle: If the wheel is rotated a distance equal to the spacing of the hinges, the wheel is identical in all respects to its initial position. Therefore it won't initiate or sustain motion.

But there's more educational fun to be had with this simple device. Let's use free-body diagrams to analyse the forces and torques on the system.

Force analysis presents no problem. The net gravitational force downward on the arms is balanced by an equal force upward exerted by the axle of the upper pulley. Also, the net downward force on the right arms equals the net downward force on the arms on the left side. There's no unblanced force on the belt.

But what about torques? It appears that they are unbalanced, since the arms on the left have greater torque about the axle of the upper pulley. Is that true, and if so, why doesn't this favor counter-clockwise motion?

## Solution.

As with so many perpetual motion device drawings, this drawing deceives us. It shows a belt running over two pulleys, but it does not show the fixed, static structure on which those pulleys must be securly attached. So we are tempted to ignore the forces on the system that are exerted on those pulleys. This seemed to present no problem with forces, for we accepted that there must be an upward force on the upper pulley to support the entire system. That force adjusts to the size necessary to support whatever the weight (within reason) of pulleys, balls, arms, belt and pulleys.

Now look at torques. Here we may fall into the trap of looking only at the torque due to gravitational forces on the balls. The balls are only part of the system; the system that we imagine (hope) will rotate consists of balls, arms belt and pulleys. No parts of the system can be neglected.

Look at the forces that have torques about the upper pulley axle. On a single extended arm with ball, there's a downward force due to gravity at the left end. The arm, descending, doesn't rotate. So what force has a torque to counter the counterclockwise torque at its left end?

We have seen that the net upward component of force on the rising and falling segments of the belt is zero. The net gravitational torque due to forces acting on those portions is also zero. We may be tempted at this point to do a messy analysis of the forces and torques at the upper and lower pulleys as the arms move across from one side to the other. Let's not, for now. We won't need to.

Look at the bigger picture, considering the movable system as a whole, balls, arms belt and pulleys. We already established that the net gravitational torque on the balls about the upper axle is clearly non-zero, and counter-clockwise. Be we ignored other things that exert torque on the system, and the biggie is: the horizontal force on the axle of the lower pulley.

Since the diagram showed no pulleys, we easily overlooked this force. We may have thought it served no function, but it is important. Imagine there were no support or anchor for the lower pulley. It, and the suspended belt would lean to the right, until the extended arms on the left belt segments droop down, and some will actually hang below and to the right of the upper axle. This horizontal force acting on the lower pulley was essential for keeping the belt in the position shown in the drawing. In fact, there would also be a downward component of force at the lower pulley to maintain the tension in the belt, to keep the extended arms from sagging down from the perpendicular position shown in the diagram.

It is that horizontal component of force on the lower pulley that is responsible for the clockwise torque (about the upper axle) that is equal and opposite to the net gravitational torque on the balls. The net torque on the moving system is zero and all is right with the universe, and physics.

Flippant answers to perpetual motion puzzles are often given without deeper understanding. Appeals to conservation laws, Stevin's principle, and even "The work done by falling weights equals the work required to lift them up again", while correct, do not reveal the underlying reasons for those principles. And they do not reveal the reasons why some people are seduced by these devices, why they think they might work, and the errors of thought that can deceive. I hope this simple example will encourage readers to look at the details more carefully. Granted, detailed analysis of some perpetual motion machines can be lengthy, tedious and even subtle, but it can be rewarding for strengthening one's unerstanding. Before glibly asserting pat answers to these puzzles, ask yourself, "Do you understand all you know about it?"

## Instructive details.

If the lower pulley weren't there, how would the belt hang? This photo gives some idea about that. A string of dice, held together with flexible tape, passes over a high quality dry ball bearing. It is delicately balanced, with 8 dies on the left and 5 on the right. All dies are identical in mass. Clearly there's more mass on the left of the axle than on the right. Yet no motion results.

The hanging portions lean to the right, as expected. Force and torque analysis of these would be interesting, but tedious, and we know the outcome. All forces acting on the system will add to zero. All forces acting on any part of the system will add to zero. Torques will add to zero for any part of the system, or for the whole system, whatever center of torques is chosen. All is right with Newton's laws.

Puzzle: how can this be?

This demo is not identical in assumed principles to Bessler's belt. There's an important difference. Bessler's belt assumes overbalanced torques, but the total mass on either side of the axle is the same. In this little demo the total mass is different on left and right and the distance of the centers of the die to the axle is the same left and right. Also, the position shown is one of unstable equilibrium. Push the dice in any direction, and the system collapses. The point of showing this photo is simply to demonstrate how such a chain leans to one side if there's no lower anchored pulley. Pay attention to such details.

• Donald Simanek, 2018, July 2019.