The Paradox Engine

[telecom]

Just want to add that in your apparatus with one disc, all the angular moments are balanced as well!

[Tusk]

does this mean that all the linear forces are balanced by the linear reactions,
and only angular moments are unbalanced, no matter which one is "free" or "paid for"?

all the angular moments are balanced as well!

Thanks telecom, I think you are now seeing 2 sets of 'force pairs',  is that correct? A simple analysis might be:

1. applied force on disk has it's reaction on the drive unit (which is essentially bench mounted to this line of force)

2. the secondary reaction at the disk axis (being unique) must be considered a special case, being essentially a reaction to the inertia of the opposite side of the disk

A more thorough analysis should probably take inertia into account for both 'force pairs'. The bench mounted drive unit resists the reaction to the applied force with the inertia of the planet; likewise the disk resists the applied force with the inertia of it's own mass.

So the full story for instance 1. is:

1. inertia of planet on drive unit : reaction to applied force on drive unit : applied force on disk : inertia of disk (total mass)

and when we apply a similar comprehensive analysis on instance 2 we get:

2. inertia of planet on drive unit : reaction to applied force on drive unit : applied force on disk : inertia of opposite side of disk X2 due lever arm : secondary reaction at disk axis : inertia of rotor arm

I think that either clears things up a little, or not; depending on your initial level of comprehension. But I would definitely call instance 2. a special case.

Importantly (if we allow the introduction of energy in a simplistic yet conventional sense) we should note that the applied force on the disk provides the energy for both motions, or acts as a mechanism of transmission of it, if you prefer. As such it must surely be clear that here at least energy has spontaneously twinned itself to motivate two distinct motions since the applied force is singular, yet double, but not halved.

Thus the paradox, and once again knowing how uncomfortable this can get I stand ready to address the more familiar concerns while the elephant quietly dances around the room lol 

[telecom]

applied force on disk : inertia of opposite side of disk X2 due lever arm : secondary reaction at disk axis : inertia of rotor arm

THis part is not absolutely clear to me.
Why it comes out as X2? opposite side?

[Tusk]

Why it comes out as X2? opposite side?

My apologies telecom, I may not have fully explained the origins of the secondary reactive force (I mentioned it in passing but not in great depth). Bear in mind that in the absence of relevant material in the literature the following explanation derives from my own experimentation and analysis, but I believe it to be correct.

In equilibrium then:

In the sketch below (and for simplicity) a rod of zero mass has two objects A and B of equal mass mounted one at each end.  A force is applied to one end (A) as shown.

The applied force must motivate both masses. Therefore the force required to motivate the opposite mass (B) is half the applied force. Thus the force of inertia opposing that force is also equal to half the applied force.

The applied force must act through the centre of mass in order to motivate the opposite side, therefore the centre of mass acts as a pivot point situated halfway between the applied force and the resistance of inertia.

Also the inertia of mass A opposes the applied force as shown, such that only half the applied force acts through the centre of mass (see blue annex in diagram).

Thus the resulting force on the 'pivot point' (centre of mass) is the sum of half the applied force and the inertial force on B, or (for simplicity) twice the force of inertia on B; and therefore equal to the applied force.

Here then we see the origins of the secondary motion which, while not in violation of CoE or CoM in it's own right yet offers up opportunities in the quest for OU. With this new 'line of code' might we not reprogram reality with a fresh outlook on the manipulation of frames of reference?

Consider that a constant force applied to a mass in equilibrium results in a constant acceleration, and since Ek = ½ mv² then by application of a constant force the kinetic energy of the mass increases exponentially; a tantilising hint of OU if we could only reduce or eliminate the cost of motivating the point of force to keep pace with the acceleration. This cost of point of force motion under normal circumstances erodes all advantage. Yet we see here in a simple peg pendulum experiment a reaction, indeed an unexpected additional force manifesting remotely from the point of applied force.

This allows us, with careful design and engineering, to apply a force from outside a frame of reference wherein the secondary reaction motivates a mass additionally to the primary motivation caused by the applied force, without the usual requirement to accelerate the point of applied force.       

[telecom]

Thus the resulting force on the 'pivot point' (centre of mass) is the sum of half the applied force and the inertial force on B, or (for simplicity) twice the force of inertia on B; and therefore equal to the applied force.


So, the rotational movement is activated by the 1 unit of the applied force, and the linear motion by the 1/2 unit of the applied force + the inertia of the mass B?
Which also totals to 1 unit of the applied force?
In this case we are getting the inertia B working for us for free? And it is equal 1/2
of the applied force?