Big try at gravity wheel

[Red_Sunset]

Hi,      we're obviously expecting our bob to end up higher than it was at the start point.
My question is: where will the bob be relevant to the pivot point, will the radius be
shorter than that of the original pendulum?                                   John.

Hi John,  "will the radius be shorter than that of the original pendulum? " I would say so, yes 
I came to a similar conclusion, the cycloid curve alteration as compared to the circle curve,  is a shortening of radius.  (I saw after posting that MarkE confirmed the same)
I need to work on a picture to show that more descriptively.

In the mean time, some thoughts on the cycloid/circular pendulum
Below are some assorted ramblings, assumptions, reasoning sequences as related to Grimer's theory & MarkE summary in very plain English.  Comments, additions, alternate views are very welcome.

Swing Radius
1.. The radius utilized by the bob in a circular path remains the same throughout the swing
2..  The radius utilized by a cycloid path can be approximated to a progressively shortening swing radius as the bob approaches it apogee, 

The Period
The swing radius is the main property that determines the period of a pendulum

Inertia
The inertia possessed by the bob is influenced mainly by the swing radius of the pendulum rather than the swing angle.
1.. A circular path having a constant radius will therefore have a constant inertia.
2.. A cycloid path having an effective changing radius would therefore have a changing inertial profile.

Path
The arc distance of the cycloid path is shorter than the circular path

Force/Distance/Energy
The force-distance of an arc path (radian path) can be translated to a torque profile.
1... The circular path torque profile follows a cos pattern since its radius is constant
2..  The cycloid path torque profile follows a reduced ~cos pattern due to a changing reducing radius at its apogee. Its initial torque profile is reduced proportionately to its reduced swing radius. Its radius lever component increasing as it separates from the cycloid template former, when at the same time the force is reducing at a cosine rate.

The crux of our interest is how we can reconcile the PE (vertical height drop) of the bob to be different between a circular and cycloid path.

It has been theorized that the inertial energy (torque profile) acquired by a cycloid path bob is less as compared to a circular path bob when released from the same height.

It has been theorized that the inertial energy (torque profile) acquired by a circular descend path bob is greater than the energy required in a cycloid ascent path to the same height (the exit height attained by the bob is higher than the entry height).

Provisional conclusion
This theorized observation leads us to believe that the proportional reduction of radius lever in the cycloid swing account for the difference of energy.   
Possible ??  The definition of different flavors/harmonics of energy is a step in the right direction ?

Open to any comment,  Red_Sunset

[MarkE]

A cycloid pendulum with a free arm length Y1 has a shorter period than a circular pendulum with the same free arm length Y1.  An ideal cycloid pendulum's period does not vary with peak swing angle, whereas the period of an ideal circular pendulum's period changes with peak swing angle.

I am waiting on Grimer to either confirm that my statement of his hypothesis is accurate, or to make any needed corrections.  A test cannot be designed until the hypothesis is explicitly understood.  Some other clarifications will also be needed such as a formula that describes the excess "third derivative energy" that Grimer contends is stored by the circular pendulum.  The reason that we need the formula is that all experiments have finite uncertainties and we need to insure that in any test the magnitude of the this extra energy that we are looking for will be big enough that it will not get buried in the experiment uncertainty.

[Grimer]

Hi,
   we're obviously expecting our bob to end up higher than it was at the start point.
My question is: where will the bob be relevant to the pivot point, will the radius be
shorter than that of the original pendulum?
                                 John.

If you go to the figure at the bottom of page 16 of this thread you will see that the pendulum
wraps itself around the chop (link below shows what a chop is).
http://www.antique-horology.org/piggott/rh/images/81v_cycloid.pdf


You will see that the pendulum is bent into a curve so obviously the bob is nearer to the original
pivot at its apogee that it was at its nadir.


You will notice I wrote the "original pivot". The pivot for the straight line section is changing. This
means we have introduced an new variable, length of pendulum shaft, into our system.


We now have three variables, NG, EG and L, the length of the shaft.


Three variable systems have the potential of transducing motion from one scale to another.


The most familiar example is the Carnot Cycle where the three variables
are volume, pressure and temperature. Motion is transmitted from the very lowest scale of temperature to the engineering scale of volume
(rotation of the pistons crankshaft) via the intermediate variable of pressure which shuttles back and forth between the two.


A less familiar example is the WhipMag Cycle where motion is transmitted from the small magnet Al spins up with his thumb to the large wheel with many magnets arranged around the radii. An example which shows the the magnetic field is not conservative.


Though why on earth nobody found this out befor Al is beyond my comprehension since we already have the magnetic refrigeration
analogue of the inverse Carnot Cycle (see diagram below).

[Grimer]

I forgot to attach my card.

[minnie]

Hi Grimer,
             loved the antique horology thing. I must admit I'm well out of my depth but am
enjoying this all the same.
     What I wanted to know was if you could build on the increase in height by reversing
the cycle. My feeling is that if the bob ends up nearer to the pivot point you could not.
   One thing that I have discovered through this topic is that is that Newton was a very
clever man.
               Thank you John.