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I would like to measure the thickness of a stack of paper without touching it. The measuring device needs to show if one sheet is removed. I do not need to know the measurement in inches. Units are not a concern here, I only need some type of reading.

What type of sensor would you recommend?

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Since you are not trying to determine the absolute height of the stack, but rather to detect a change, I'd suggest not measuring thickness at all. Nominal thickness of 20 lb paper is .004 inches. You could use essentially a microscope with a video camera, and adjust the height to provide maximum maximum contrast in the image. This is basically how cameras autofocus. You'd need a microscope lens to get the depth of field down to thousandths of an inch - regular camera lenses won't do the trick.

But, as I say, I'd recommend a different approach. Specifically, I'd go with weighing the stack, then detecting when the weight has decreased. A stack of 100 pieces of paper will decrease in weight by 1% when one sheet is removed, and that ought to be much easier to detect.

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Typical paper is pretty thin (a few thousandths of an inch or less). Using thicker paper would make the problem easier, but assuming you can't do that..

You could use an LVDT with a spring on the core to provide downward pressure and a puck at the other end with an air nozzle or porous surface supplied with compressed air to prevent the puck from touching the paper when it gets very close.

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Can you weight it? If there are not too many pieces you could look at transmission at some IR wavelength. (We use to stack up paper for IR attenuation in grad school.)

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