How to Calculate what is left on a roll of say wallpaper without unrolling it.

Carlow52

It is actually a roll of breathable membrane.
I know

• the original diameter is, say D
• the diameter of what is left, say d.
• the original length, say L

The cardboard core is 30mm in diameter.
Thanks

red_fox

Carlow52 wrote: »

It is actually a roll of breathable membrane.
I know

• the original diameter is, say D
• the diameter of what is left, say d.
• the original length, say L

The cardboard core is 30mm in diameter.
Thanks

The length will be proportional to the area of the side, so

[(d^2 - (30mm)^2)/(D^2-(30mm)^2)]*L

would be your answer, I've dropped pi and that fact that d = r/2 because they get canceled anyway.

But practically I think it would be more accurate if you had a good scales, a full roll, a cardboard tube and the roll in question, weighted the three and did similar

(Weight of roll X - weight of tube)*L

---

Weight of full roll - weight of tube


but that's only because there may be less margin for error in your measurements.

gondorff

In layman's terms:

([pi(rtotal)^2 minus pi(rcardboardroll)^2] / thickness of material)(sqrt cross sectional area)

A little convoluted? Dunno, but I think it works.

alan4cult

Wow, cool problem.

Well if you rolled out the roll when you first got it it would be "L" long and "t" thick.

So the cross sectional area would be L*t.
If you rolled it back on again (really tight!) the cross sectional area would be measured by:

pi(R)^2 - pi(r)^2
Where R is the outer radius and r the inner radius.

Now
L*t = pi(R)^2 - pi(r)^2
L*t = pi(R^2 - r^2)

So simply if you want to find what's left on the roll

L = pi(R^2 - r^2) divided by thickness

Or you could use this:
http://www.cutsmart.com/pages/roll-length.html

gondorff

Spot on.

I think this is what I was trying to say - if you disregard the (sqrt cross sectional area) bit... dunno what I was thinking with that one.

MathsManiac

That's all very well, but if you look at the original post, you'll see that the thickness of the membrane is not known. In reality, it would also be difficult to measure accurately.

Given the information that IS available, I believe red_fox's answer is more practical.

RoundTower

MathsManiac wrote: »

That's all very well, but if you look at the original post, you'll see that the thickness of the membrane is not known. In reality, it would also be difficult to measure accurately.

Given the information that IS available, I believe red_fox's answer is more practical.

In practice I think weighing it will be more accurate. Especially if the material is heavy and compressible, or on the other hand if it is easily deformed so that the roll it won't have a circular cross section.

LeixlipRed

I'm no physicist (apparently this is what they do all day ) but I reckon weight would be a better approximation as well. Obviously some wallpaper isn't of uniform thickness so this would affect your calculations more on the cross section type approach I reckon.

MathsManiac

I agree. But, as red_fox pointed out, to do that you need:

red_fox wrote: »

a good scales, a full roll, a cardboard tube and the roll in question

So unless you have these things lying around, or you had the presence of mind to weigh the full roll before you started, and weigh the tube from the last roll you used before you threw it away...

RoundTower

MathsManiac wrote: »

I agree. But, as red_fox pointed out, to do that you need:

So unless you have these things lying around, or you had the presence of mind to weigh the full roll before you started, and weigh the tube from the last roll you used before you threw it away...

well the diameter way you need to know the original diameter of the roll. Of course this was specified in the question, but in practice it's not much more likely than knowing the weight.

MathsManiac

RoundTower wrote: »

well the diameter way you need to know the original diameter of the roll. Of course this was specified in the question, but in practice it's not much more likely than knowing the weight.

Good point. So, (s)he probably has a new roll available (with a label on it telling him the total length, I guess). Empty roll is the only remaining problem...

RoundTower

of course, you could set the roll spinning around the tube and calculate its angular velocity.