## Belt and Pulley Devices, the Simple Answer.

The "logic" of the inventor of these devices goes something like this:

• Make the weights fall or move so that they have a greater lever arm on one side of the belt.
• Greater torque on one side of the belt will cause continual cyclic motion. (This is the clinker in this argument.)
• If more power is needed (say, to overcome friction!) just make the belt longer and add more weights to it so there's more torque imbalance.
If this seems at all persuasive to you, you've taken the first step toward self-delusion, and you are more likely to overlook the real flaw in this device. Several correspondents noted the practical point that the weighted rods fastened to the belt will cause the belt to buckle a bit where they are attached, and thought if this could somehow be eliminated, the device would work as a perpetual motion machine. Vain hope.

The device does seem to have torque imbalance due to the continually greater lever arms on the right. Weight imbalance is "iffy", there's eight balls on either side of the central axis. Maybe we should look at the forces that act on the belt.

Consider the vertical sections of belt. Each of these has 5 rods and 5 sliding weights. These weights are responsible for the forces acting on the belt where the rods are attached. These forces are equal on the left belt segment and the right belt segment. Therefore the upper pulley supports the same weight on its left and right sides, and there's no net torque to turn the pulley. This is true no matter where each weight is on its rod. End of story.

But you still may be bothered about the apparent torque imbalance that may have initially distracted you from seeing this solution. You need to do a proper force and torque analysis using free-body diagrams.

Consider a belt made up of short rigid sections linked together, with a rod rigidly attached to each, and a sliding weight on the rod. Now, let's break this system down by looking at just two of these, one on the side of the belt which allows the weight to slide to the far end of the rod well away from the belt, and one on the other side, where the weight rests near the belt. [Fig 2-1] The belt links tilt, and the gravitational torque on the weight is balanced by the couple due to the belt tension. [Fig 2-2] The belt sections above and below the link are now a bit off vertical, and this causes horizontal comonents of force at the upper and lower pulleys, where the belt contacts the pulleys. [Fig 2-3] And, in turn, the pulley bearings counter these forces with two horizontal forces, forming a couple. But remember, horizontal forces at the bearings cannot cause the pulleys to rotate. So there is more torque on the right side, but this is balanced by the torque due to horizontal forces at the pulley bearings. The net torque on the system is zero. [Fig 2-4]

This is the process of analyzing a system by considering free-body force diagrams on each of its parts. We have shown only the parts we need to examine, and the figures are schematic only, the forces not being drawn to scale. In particular, we have greatly exagerrated the size of the kink in the belt.

• Fig. 2-1. The idealized system.
• Fig. 2-2. Showing the kink, much exaggeratted. Only forces on the kinked portion are shown. These three vectors must sum to zero. The torques on this portion also add to zero because of the couple formed by the two tension forces.
• Fig. 2-3. The forces acting on the movable part of the upper pulley are shown. V and H are the vertical and horizontal components of the bearing force on the pulley. The downward force T2 has the same magnitude as the upward force T2 of fig. 1-2, by Newton's third law.
• The force T2 has components H and T. The two forces labeled H are equal and oppositely directed because they are the only two forces on the pulley and as the pulley does not move horizontally, the net horizontal force on it must be zero.
• Fig. 2-4] shows only force W and the two forces H which provide torques that add to zero, and these are the key to understanding why this system has no tendency to turn. All the other forces also have torques that add to zero as well.
In fact, the downward component of force at the upper pulley is the same whether the weights are near the belt, or at the end of the rods farther from the belt. As the weights shift along the rods, the vertical components of force on the belt do not change, so this does not initiatae or sustain motion of the belt. The weight supported by the upper pulley is the same no matter where the balls are. The system is not dynamically or statically unbalanced by the sliding weights. This demonstrates the role of internal contact reaction forces in a system made up of several parts.

This analysis considered just one sliding weight. In a belt with many of them things get a bit more complicated to draw, but the conclusion is the same. If the belt with one weight has no tendency to turn, the belt with N weights won't turn either.

Once you do this analysis, thinking through the force and torque analysis, you then wonder "How could this have ever looked the least bit plausible as a perpetual motion machine?" Then you realize how seductive these devices can be when only casually considered.

It also becomes clear that this pulley device (and most other belt and pulley perpetual motion machines) are nothing but stretched versions of the early overbalanced wheels, which also didn't work. Adding the belts gains nothing but additional friction.

But, to the average person there's something persuasive about the belt, and the fact that on the vertical segments the system obviously has weights continually extending out farther from the pulley axle. This leads inexperienced people to suppose that that there is a torque imbalance that must cause the system to turn continually.

The experienced engineer or physicist goes right to the simplest and most reliable method of analysis. Look at the work done as the weights move vertically. The weights are equally spaced along the belt, so there are always the same number of them on either side of the system. And in a given time, the number moving up equals the number moving down, and they move the same distance. So the net work in moving the system is zero in any given time. The distance of these weights from the axles is simply irrelevant, since whatever this distance the weights move the same distance vertically. This also allows us to conclude that lengthening the belt won't help at all, for if N is the number of balls, N×0 = 0 whatever the size of N.

Thought problem. Has this any functional similarity to the Roberval balance?