The pendulum bias paradox experiment

[TinselKoala]

Because momentum is conserved. You apparently want to look for conservation of energy in the wrong place. I have told you already that CofM accounts for the center of mass remaining in the same place, and that CofE accounts for each ball rebounding to its original height. Assuming perfectly elastic collisions of course. There is no paradox, simply a misapplication, apparently, by you, of the correct laws in the correct place. Again, I believe that this is deliberate and that you have some agenda down the line and you are leading up to something. So get to it.

It would be a neat trick to get your "perfectly inelastic" clay balls to stop dead when they hit; this would require the _total_ kinetic energy of the two balls to be completely dissipated as deformation, heating, etc. perfectly symmetrically throughout the resulting mass. In a thought experiment, maybe. In the real world, I doubt it. The internet is full of examples of partially damped "inertial thrusters" that  "work" by partitioning kinetic energy unequally this way and thus, moving their centers of mass, in a manner that you apparently might call a paradox. I've made a few myself.

If you want me to explain to you why momentum is conserved, or why _total_ energy is conserved..... the best I can do for you there is to appeal to an anthropic principle of sorts: because if the Universe were otherwise, it would not look like it does now and we wouldn't be here to view it. Or I could refer you to God ... but he's not talking, at least not to me.

[Pirate88179]

Why does the ball with greater kinetic energy not disturb the centre of mass of the system, which is static throughout the experiment?

I take exception to this conclusion.  The center of mass does indeed move, just enough to obtain equalibrium  which, nature seems to like.  If studied closely using a rigid test device, this would become obvious.

Bill

[TinselKoala]

I take exception to this conclusion.  The center of mass does indeed move, just enough to obtain equalibrium  which, nature seems to like.  If studied closely using a rigid test device, this would become obvious.

Bill

I think you are right too, Bill. I think if you idealize the masses to points and have their suspensions at the same point, ideally the center of mass wouldn't move. But this is impossible to achieve in reality with balls of different diameters and suspensions separated by a finite distance.

[MileHigh]

I haven't read the last few days but just skimmed the last few postings.

Some comments that may help:

Conservation of momentum typically is discussed in cases of an inelastic collision.  Momentum is conserved and energy is lost, but often energy is not even being considered in cases like this, just momentum.

When the balls are about to hit, and after the hit,  it would appear the center of mass for the two balls does not change, and the center of mass is not moving.  You observe a nearly perfectly elastic collision.  It's like each individual ball hit a hard wall and bounced off.  They must have equal kinetic energy when they hit for that to happen.  If there was an imbalance in kinetic energy then the center of mass between the two balls would have to be moving after the collision and it doesn't.  You would notice an asymmetry in the the behaviour of the balls with the first few successive hits and you don't see any.

So you are left with both the small and the large ball hitting the block of wood with the same kinetic energy, but different masses and velocities.  The block of wood reacts appropriately for each type of hit as was previously explained.

[Tusk]

A fair attempt TK.

Because momentum is conserved

CofM accounts for the center of mass remaining in the same place

CofE accounts for each ball rebounding to its original height

As I stated earlier, not recognising the paradox merely indicates an 'off the shelf' perception based in convention.

If you want me to explain to you why momentum is conserved, or why _total_ energy is conserved.....

Clearly you cannot, neither the vast majority. But your perception of the phenomena is based completely on established laws with no real understanding of the 'how' or 'why'. No shame in that but since you seem unlikely to acknowledge the difference, turning over new ground will be all but impossible, not unlike investigating gravity with someone completely comfortable with the idea of things somehow magically 'pulling at each other' across the void (Mr Einstein's warped fishnet of space-time notwithstanding). This with no rancour, we are at odds merely due to disparate perspectives.

So, I will give you the answer, although at this point for what purpose I am not certain. Why does the ball with greater kinetic energy not dominate?

As a first step let's examine How:

Throughout the collision the forces applied by each ball to the other must be equal, and while these forces vary throughout the collision we can for the sake of clarity treat them as constant. Since kinetic energy is a function of velocity it is also a function of distance. The centre of mass of the ball with the greater kinetic energy must therefore travel a greater distance during the collision because the period during which the collision takes place is the same for both balls (the acceleration of the balls cannot be equal as this would produce unequal forces). Thus both balls come to a state of rest each having applied an equal force to the other for an equal period of time.

Why?

Since all motion (and every collision) is governed by momentum therefore kinetic energy is simply an artifact of the frame of reference in which it is observed. If our frame of reference were the centre of mass of the large ball (pre-collision) we would observe that the small ball had high kinetic energy and post collision results would be in accordance with that observation. If our frame of reference were the centre of mass of the small ball then it would have no kinetic energy. But since our frame of reference for the experiment is the centre of mass of the system we observe the true nature of the phenomena, determined solely by momentum with the kinetic energy of the balls seemingly unequal yet in fact simply not relevant. Kinetic energy only manifests when measured against a 'static' point in our frame of reference, or in other words when we place ourselves in the same frame of reference as 'the large ball'. The reality and determination of every collision resides in the domain of momentum.

Conclusions

The experiment demonstrates what is already known; kinetic energy is not invariant. I use the double negative here just as it often appears in the literature. If you want to say something without anyone really hearing it use a double negative. So we could also say that kinetic energy is variable (depending on frame of reference). It follows that conservation of energy, at least with reference to kinetic energy, is also frame of reference dependent.

As a result, anyone interested in mechanical overunity might want to think seriously about the mechanical manipulation of frames of reference, without which conservation of (kinetic) energy dictates that there can be no advantage.

As stated previously there is more, but perhaps this material is a tad too spicy for the local palate.