I need to know how fast in RPM to rotate the core of the Paper Towel to keep up with the feeding system. Keep in mind that the diameter of my Paper Towels on the roll will be growing in size and as the diameter increases my RPM's must drop.

100 FPMto RPM

I have looked here (and the second link seems more relevant but I am not a math wizard) : Speed calculation with Diameter and RPM

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Feet perminute Calculator

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100 FPM feed, and a Circumference of 10 feet , would mean I need 10 RPM to match Feed speed (each revolution I understand will have a different rate as the diameter grows so therefore does my circumference ).

The linear speed at the edge of the spool it is related to the angular frequency by $$v=\omega r$$ Here $r=d/2$ is the radius, $d$ is the diameter. The angular frequency $\omega$ tells you how many radians the spool rotates in one second. You can transform this into revolutions per second by diving $\omega$ by $2\pi$, then to RPM by multiplying the answer by $60$ (seconds/minute). So $$v=\frac{60 RPM}{2\pi}\frac{d}{2}$$ or $$RPM=\frac{\pi v}{15d}$$ If you measure the velocity in ft/min, you must also measure the diameter in ft.

1750RPM toFPM

Feet per min to rpmconversion chart

I understand that the RPM will vary as the roll gets thicker - my point exactly in needing to know this value. Imagine doing this calculation at undetermined intervals of say 100ms +/- 20ms.

Would RPM of the takeup spool = [(FeetPerMinute of the feeding system) / (Circumference of Takeup Spool) ] Where Time Base is in minutes and Circumference is in Feet ?

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I do not have a TIME variable as the calculation device has no accurate time base to use for calculation (however Feed rate is in Feet Per Minute).

I have the center core measure of 4". Each time I calculate this I will have the diameter value which also gives me the Delta of Diameter. I can't guarantee thickness nor can I guarantee that I will have less than 1 wrap of the core each time I measure the diameter, although it is probably true enough to say it will be some where less than 1 wrap at each measure cycle.

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Feet per min to rpmcalculator

and here: How do I calculate the instantaneous angular speed of a spindle given a spooled radius and required feed speed?

I need a formula for finding out what RPM I need to rotate a spool of material in order to keep up with a given feed rate supplying that spool.