Can energy be extracted indefinitely from a mass dipole/mass negative mass pair in perpetual motion?

Question

Some days ago I have read this short piece of article by Hammond which caused me to revisit the theoretical properties of negative mass.

It is known that a pair consists of ordinary mass and negative mass of equal magnitude, known as a mass dipole in some sources, can accelerate indefinitely by itself while the total energy of the system remains zero at all times, thus result in a bizarre form of perpetual motion.

I then tried to model a system where the mass dipole is traveling in circular motion (the mass dipole is brought into motion by nudging one of the masses with an initial velocity $v_{1}$ so as to break the unstable equilibrium the mass dipole is at, assuming the rod attached to the mass is rigid) and see if energy can be extracted from it without the system grind to a halt (as that is always the case for any classical perpetual motion generator case studies such as the water screw).

Using the free body diagram for each of the masses as shown, and the following assumptions:

1. Assume both masses travel with $v_1$ at $t=0$
2. Assume the dynamo has an efficiency $\eta$ and it converted rotational kinetic energy to electrical energy
3. Assume anything else in the system that is not the negative mass is made of ordinary mass

Q1. What other assumptions I have missing or is wrong in order to model this more realistically (in particular, will the centripetal force on the negative mass causes the system to fly apart eventually due to how for negative mass F=-ma)?

After the antipodal configuration of the mass dipole is being ruled out of perpetual motion generator. A more general scenario is considered as follows:

Here, to remove the notion of up/down, the whole set up is laid horizontally on a plane. Due to the closely spaced mass dipole of distance $\lambda$ away, they both experienced a gravitational force and corresponding acceleration radially towards each other as shown by $F_{g2}$ and $F_{g1}$ where $|F_{g1}|=|F_{g2}|=F_g$. The acceleration of the negative (inertial) mass ($a_{g2}$) is opposite to the direction of the force as shown.

By trigonometry where the angles $\theta$ and $\phi$ were highlighted, it is easy to see both accelerations resolve into tangential and radial components of the same magnitude (one of these, $F_{c1}(0)$ is illustrated for $t=0$). Since the rods are rigid (but can move independent of each other as they are attached to bearings with friction $f_1$ and $f_2$ respectively) a nonuniform circular motion is expected. The geometry of the setup also means the acceleration hence the tangential acceleration will have fixed magnitude as the mass dipole revolve around the setup, carrying the rods with it that attaches to the dynamo.

If the friction of the bearing is made to almost compensate for the tangential acceleration i.e. $|f_1+f_2| +\delta F = F_g \cos \phi$, where $\delta F$ is a tiny amount of force, then as the dynamo extract work, hence by work energy theorem, the mass dipole will slow down during the extraction as the kinetic energy of each mass decreases (despite the total kinetic energy remains zero). Then, as long the velocity of the setup does not slow beyond a threshold which result in the friction to overcome the acceleration and halt the motion, then theoretically the set up will allow unlimited extraction of energy since the total energy of the system remains zero, and that once the extraction is halted, it will accelerate again due to the net force on the mass dipole.

It is unclear how entropy should be modelled for this system. It is clear due to the restriction of the rod, the configuration that the system can visit is fixed. However the energy of the mass dipole is zero, as there is no potential and the total kinetic energy is also zero. The overall entropy of the combined dynamo and mass dipole system seemed to be increasing (due to the inefficiency of the bearings and the dynamo itself), thus the perpetual motion generator does not seemed to violate the second law of thermodynamics, yet it is inexhaustible as it can always accelerate again when the dynamo was not turning.

Q1. It is well known that all perpetual motion generators violates the second law of thermodynamics. What are factors are needed to better model the problem. Is this system actually act as a perpetual motion generator obeying the second law of thermodynamics?

Q2. How to model the emission of black body radiation for a negative mass?

Q3. Any other implications for this system besides the perpetual motion property?

Answer 1

Your mistake is here:

3. Assume the gravitational force acts with a distance of $πr$ due to how the rod supporting the mass is rigid thus restricting them from interacting directly with the separation $2r$

There's nothing magical about a rigid rod that re-directs the force of gravity. The gravitational force between your two masses will be radial, not tangential, with a separation of $2r$. Since this force passes through the axis of rotation, there is no net rotational acceleration, and friction will dissipate the initial rotational velocity.