# Finding thickness of paper based on rate of change of radius and constant speed

1. Sep 6, 2011

### Jaboska

I need find the thickness of a piece of paper based on the known linear speed of the paper coming off the roll and the change in diameter of the roll.

I have the ability to see the linear speed. It is consistant but changes from roll to roll. I know the starting radius, instantaneous radius and ending radius. The thickness of the paper is consistant but changes from roll to roll.

The application for this is stopping the roll of paper with only a 1/4 of an inch left on it with a starting radius of about 24 in and a constant speed varring between 70 to 300 m/min.

I already have the formulas to calculate when to begin deceleration, but I need know the thickness of the paper based only of the rate of change of the radius.

Thank you for your help

2. Sep 6, 2011

### Staff: Mentor

Does the speed vary only when the roll changes, or does it vary also on the same roll?

3. Sep 6, 2011

### Ken G

You can do it by equating changes in volume. The rate the volume is leaving, for an instantaneous radius r, is d/dt of pi*r2*w, where w is the (irrelevant) width of the paper. The rate the volume is coming off is w*x*v, where v is the speed and x is what you want. So we have:
pi*dr2/dt = v*x.
All you need to know is the ratio of dr2/dt over v, at any given time, and you can infer x. If you can't measure dr2/dt instantaneously, you have to infer it over a finite time, like delta r2 over delta t, and that will be right because it doesn't change if v and x are constant.

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