Buoyant Optimism

One test of physics understanding is to use it to examine unusual devices and even unworkable devices. Here's a classic perpetual motion proposal.

A cylinder (or sphere), A, on a fixed axle has a tank of water, D, surrounding its left side. A frictionless watertight seal prevents water leakage. We also simplify the problem by using a frictionless axle for the cylinder.

The inventor says the physics is sound, textbook physics. Archimedes' principle says that a body fully or partially immersed in liquid experiences an upward buoyant force equal to the weight of the displaced liquid. Clearly the left half of the cylinder displaces liquid, so it should experience a buoyant force upward, causing a torque that will rotate the drum clockwise. At least that's the inventor's argument.

Now I assume all readers immediatly conclude that this will not work, and the drum will not turn. But can you explain why?

One often hears the lame observation that "Perpetual motion devices are impossible because you can never eliminate friction and other energy dissipative processes." But the reason such devices are interesting as physics puzzles is that even if all dissipative effects are removed, they still don't work. So those energy wasting processes are not the reason such devices cannot work. We have deflected that response by giving this device a frictionless axle and leakproof, frictionless seals. Idealized, yes, but the puzzle is still instructive.

The lazy person's argument is "It won't work because it would violate the laws of thermodynamics and conservation of energy." A more sophisticated argument notes that any virtual (imagined) rotation of the drum through any angle takes the system from one state to another, and both states are identical. Both states look the same, have the same distribution of mass and all parts of the system have the same center of mass in both states. So by Stevin's principle of virtual work, the drum will not move from one state to another without external energy input.

These arguments from general principles are valid. But they sidestep the physical details of forces and torques. A free-body analysis of forces and torques would be a good exercise.

But what about the buoyant force argument? Is there a buoyant force on the drum? Will it exert a torque on the drum? If so, why doesn't the drum rotate? Or doesn't Archimedes' principle apply here?

Answer

But let's look at it another way. Is there a buoyant force on the drum? If so, why doesn't it initiate and sustain motion of the drum?

The sum of the vertical components of force of water acting on the drum is indeed a buoyant force, acting upward through the center of buoyancy. It exerts a torque on the drum, clockwise in this case. But we already concluded that the net torque around the drum's axle must be zero, so there must be another torque somewhere. One of the commonest errors of perpetual motion machine inventors is to overlook some force or torque. Where is this other torque? The frictionless axle bearing cannot supply it.

If a force has zero torque about an axis, its components will have torques that add to zero about that axis. In two dimensions, that means that the two cartesian components of the sum have equal and opposite torques abuout that axis.

We have established that the net force due to pressure on the drum has no torque. The torque due to its vertical components must be balanced by an equal and opposite torque due to the horizonal components.

Look at the horizontal components of force due to pressure acting on the drum. The size of these horizontal components increases with depth, so the sum of the components below the axle is greater than those above the axle. Their counter-clockwise torque on the drum is greater than the clockwise torque of those above the axle. The net counter-clockwise torque due to horizontal components is exactly equal and opposite to the clockwise torque due to the vertical components. The net torque on drum is zero, as we expected. If you doubt this, do the vector calculus.

All introductory textbook examples I've ever seen dealing with buoyant force have the immersed portion of the solid body surrounded by liquid on all sides and below. The horizontal components of pressure forces on the immersed body add to zero. We get so accustomed to this kind of situation that when we encounter an example like our puzzle, where the horizontal components do not add to zero, we carelessly ignore them. That makes this unworkable design especially interesting and instructive for students.

What important principle does this teach us? It is simply this: "Nature abhors perpetual motion." I.e., nature does not allow perpetual motion. From this principle we can derive, as a consequence, the laws of conservation of energy, momentum, angular momentum and even Newton's laws.

Many other unworkable devices are presented as puzzles on my web pages The Museum of Unworkable Devices. My email inbox continually receives descriptions of such devices from optimistic inventors, mostly just variations of classic historic ideas, even some that have been patented. Restless minds continue to re-invent square wheels. Some have baroque complexity. But none are as good as a simple flywheel on frictionless bearings. Whenever the inventor adds gimmicks to a simple wheel, like moving weights, gears, articulated arms, liquids, or magnets, its performance is made worse.

Reader feedback is appreciated. If you have a favorite physics puzzle that is not well known, not easily found on the web, or in the many published physics problem books, send it along. Include your answer, if you have one. If your idea is used, we'll credit you if you wish. I especially like puzzles that can be solved with insightful and simple arguments, preferably with minimal mathematics.