Analysis of nmgolfer's ideas about what makes the bead slide of the rod.

nmgolfer's ideas are clearly exposed in his article above. They are summarized below.  

-1- No resultant force acts on the bead when the restrained is eliminated at t=0.
-2- Henceforth, the bead slides off the rod, per Newton's first law, in a straight line.
-3- No centrifugal force is involved.

We have taken exactly the same situation as used by nmgolfer - constant angular velocity - and analyzed it carefully.

The governing equation of motion for this particular case are readily arrived,

The only external force acting on the bead is the reaction force F exerted by the rod on the bead.  It is equal and opposite in value and balancing out the Coriolis force, acting on the bead.

From relation (1) it is evident that there is indeed a centrifugal force acting on the bead. This contradicts totally with nmgolfer's assertion.

The general solution of equation (1) is:

Constants C1 and C2 are determined respectively by the initial values r[0] and  r'[0].  We have taken, as nmgolfer, r'[0]=0 and assume r[0] to be equal to 1. This gives C1=1 and C2=0.

Fig1

Fig1 shows relation (4) in a more convenient graphical form. Also shown is the vector representation of the centrifugal force acting on the bead.  nmgolfer vigorously asserts however that it plays no role whatsoever.

Fig1 shows clearly that nmgolfer's assertion about the straight line motion of the bead sliding towards the end of the rod doesn't hold any water.  Moreover he also has to revise his ideas about centrifugal forces.

Fig2

Fig2 shows the three forces acting on the bead whilst sliding off the rotating rod. One force is the reaction force the rod exerts upon the bead. In this particular case, with constant angular speed, there should be no resultant tangential force acting on the bead.

This is indeed the case with the Coriolis force and the rod reaction force being equal and opposite and hence balancing. The only net force acting on the bead is the centrifugal force acting outwardly along the rotating rod.

For those being intrigued by nmgolfer's strong denial of centrifugal force I propose a simple thought experiment. Look up in the dictionary the word 'accelerometer'. It will tell you that it is a device used to measure force.

If one can measure something we will be definitely forced to admit its existence, that is inherent in a logical common sense approach necessary and required in any scientific endeavour.

So let's imagine to purchase two micro wireless accelerometers and we subsequently glue them on the bead, taking care to align their measuring axes respectively parallel to the rod and perpendicular to the rod.

The measurements obtained will be as shown in Fig2, ie., zero force perpendicular to the rod and a net varying centrifugal force, corresponding to the colored arrow, along the rod.

Everything went fine, nice experimental results. Yes, but there is only one small nasty problem. nmgolfer comes around and tells you that what you so diligently measured does not really exist.

You did probably not say it, being impressed by his impressive credentials, but you silently wondered if science is always so inconsequential. You measured patiently and accurately something and get told that it does not exist.

You start wondering indeed to take him very serious since, in your experiments, you noticed a interesting curvilinear trajectory yet he tells you that this is just your imagination, it is a straight line trajectory.

Conclusion

nmgolfer's claims:

-1- No resultant force acts on the bead when the restrained is eliminated at t=0.
-2- Henceforth, the bead slides off the rod, per Newton's first law, in a straight line.
-3- No centrifugal force is involved.

mandrin's conclusions:

-1- There is a resultant centrifugal force exerted on the bead.
-2- The bead does not slide off the rod in a straight line.
-3- Centrifugal force is involved.

Anyone can readily judge for himself. I started of with nmgolfer's ideas and followed up with a mathematical analysis, proving nmgolfer dead wrong with his assertions.

mandrin