Transfer Momentum Along Axis by Torque: Special Relativity Analysis

[Jonathan Scott]

If you spin up a flywheel by applying torque somewhere else on its axle, you are transferring energy along the axle from the point where the torque is being applied to the flywheel, which effectively adds to the mass of the flywheel. This means rather counter-intuitively that a tiny force has to be applied along the line of the axle to keep it from shifting slightly towards the point where the energy is being applied (by an amount such that the center of mass of the added energy and the original rest energy remains at rest).

I analyzed this system many years ago in response to a posting by a Dr. Eue Jin Jeong, who had not spotted this possibility and instead thought that this was a means of shifting the center of mass of the system in violation of conservation laws, which would lead to what he calls "Dipole Gravity". (He ignored my analysis and carried on by writing books about it). I think I assumed a hollow tube for an axle and a structure built of narrow isosceles triangles carrying the forces along it, but I no longer have my notes from that time, and I don't have the patience to reconstruct the analysis at the moment.

Does anyone know of a suitable reference for the way in which Special Relativity means that any sideways transfer of force (especially as part of a torque) also implies a sideways component of force corresponding to the sideways rate of transfer of energy?

 

[Jonathan Scott]

I guess that if I want the answer I'll have to do the analysis myself. It's clear that if energy is "pushed" along an axle by a torque, it takes a tiny force along the line of the axle to get it moving then an opposing force at the target location to bring it to a halt, in and that if those forces are not opposed (for example by stops at the ends of the axle) then the axle would shift in such a way as to keep the overall center of energy at rest. So even though energy may be supplied via a torque which is not intended to cause any forces in the axial direction, the theoretical result involves forces wherever energy is being moved.

After a bit of searching I found part of my previous discussions in sci.physics dated 18th October 1995 (so it's not surprising I couldn't remember much about it), in which I suggested that light speed delays mean that regardless of how rigid the axle might be, a torque will always cause a slight helical twist which will cause forces along the length, but I can't find anything more detailed.

It may be that I never managed a convincing microscopic explanation, even though Special Relativity and conservation laws mean there must be one.

 

[jartsa]

Does anyone know of a suitable reference for the way in which Special Relativity means that any sideways transfer of force (especially as part of a torque) also implies a sideways component of force corresponding to the sideways rate of transfer of energy?

Sideways transfer of force does not imply axial force. Because when there is a constant force and a constant flow of energy, there is no net acceleration of energy, and no net reaction force caused by acceleration of energy.

There is a net acceleration of energy when some energy is launched towards the destination, but no energy has yet reached the destination.

EDIT: Oh, maybe it's the force on the energy sending or energy receiving end that we want to know?

 

[Jonathan Scott]

There isn't an overall force in this case, but there is effectively a time delay between the energy being launched (which requires an impulse) and the energy being brought to a halt at its destination (with a matching opposite impulse). That means that if the system is free to move, the launch sets it slightly in motion and bringing the energy to a halt stops it again, but there is an overall shift of the center of mass according to the amount of energy transferred.

The speed at which the energy is sent cancels out of the result, as the slower the speed, the longer it is in flight by the same factor.

The overall macroscopic picture is quite straightforward combining Special Relativity and Newtonian conservation rules. What I'd like is a quantitative microscopic picture which explains how applying a torque to an axle in such a way as to transfer energy necessarily results in a tiny force component along the line of the axle.

 

[jartsa]

The overall macroscopic picture is quite straightforward combining Special Relativity and Newtonian conservation rules. What I'd like is a quantitative microscopic picture which explains how applying a torque to an axle in such a way as to transfer energy necessarily results in a tiny force component along the line of the axle.

If we go just to the level of one rod, which is transferring a push or a pull force, while being a part of an energy transferring structure, the direction of energy transfer at some moment becomes a frame-dependent thing. For example in a frame where the rod stays momentarily still there is no energy transfer at all at that moment.

So, I guess we only need to understand a moving rod that is being pushed from both ends by forces that have the same magnitude. Sounds quite simple

 

[Jonathan Scott]

There is no problem explaining how the system behaves at a macroscopic level. Energy is being added at one place and removed at another, and the energy flow is associated with a corresponding transfer of momentum along the direction of flow, resulting in a shift of the center of mass by an amount corresponding to the added energy. The effect is of course too tiny to measure in any mechanical system, but is required by conservation laws.

I'd just like to be able to explain how at the microscopic scale a torque which should apparently have no component in the direction of the axle results in a theoretical tiny reaction force along the axle which corresponds to the impulse needed to send the energy on its way.

 

[jartsa]

There is no problem explaining how the system behaves at a macroscopic level. Energy is being added at one place and removed at another, and the energy flow is associated with a corresponding transfer of momentum along the direction of flow, resulting in a shift of the center of mass by an amount corresponding to the added energy. The effect is of course too tiny to measure in any mechanical system, but is required by conservation laws.

I'd just like to be able to explain how at the microscopic scale a torque which should apparently have no component in the direction of the axle results in a theoretical tiny reaction force along the axle which corresponds to the impulse needed to send the energy on its way.

I understood that, so I was trying to get down to the microscopic level, so I imagined a rigid structure made of many rods, but at the rod level I got the impression that I would not find anything new at smaller level, because I would be studying a small part of a rod under tension, which is not fundamentally different to a larger part of the same rod.

At the most macroscopic level we have energy that experiences some accelerations, in all frames. At rod-level, in one frame energy flows through a bent rod, so the energy accelerates when flowing through the bend, but in another frame energy is not flowing through that same rod, so there's no acceleration of energy at the bend, and no force that causes the energy to accelerate.

 

[Jonathan Scott]

I think I previously had some success with reducing the axle to a hollow tube made of meshes of triangles of springy rods which can only support pushing or pulling. Note that transmitting a force laterally via such a mechanism involves pushing one side of the triangle and pulling the other. However, I don't recall the details or even whether I succeeded in making such a model work. I remember I also applied it to a simpler case of accelerating an object linearly using a lateral strut attached to a trolley being pushed along a rail in a straight line, and I think I managed to get that one to make sense, although I think that one involved a simultaneity gradient.

 

[jartsa]

How about if we consider two massive disks rotating with the same angular velocity, and with opposite point charges at opposing positions on the two disks.

When we start gently braking one disk, electro-magnetic forces limit the relative angular displacement of the disks, just like electro-magnetic forces limit the twisting of an energy transferring axle.

If the disks are far from each other we can say that one disk is an energy emitting antenna that is experiencing radiation pressure, while the other disk is a energy absorbing antenna that is also experiencing radiation pressure. When the braking starts, one of the disks starts experiencing the radiation pressure a little bit earlier that the other one, so the center of mass of the two disks is displaced very slightly.

If the disks are close to each other ... I don't know, but maybe there is still a radiation pressure, which is maybe proportional to the poynting vector. Wikipedia seems to be saying something like that: https://en.wikipedia.org/wiki/Poynting_vector#Radiation_pressure

 

[Jonathan Scott]

How about if we consider two massive disks rotating with the same angular velocity, and with opposite point charges at opposing positions on the two disks. ...

I don't see enough similarity to my hypothetical "mechanical" system for that to help.
I've received other hints that (as mentioned in my second post in this thread) it may help to apply Lorentz transformations to a typical element of the axle at the time that the angular acceleration is in progress, which may well result in a component of force in another direction. I'll update this thread if I find a way to do that which makes sense.