Measurement Challenge

Wrap the note around the area to be measured [ you may have to tear it into a strip, so it fits between the flares], mark with pencil @ overlap, put note flat on desk, Measure w/thick steel rule.÷by pi find sq root x 2. Hold up rule & eyeball [to make sure your answer makes sense] Accuracy ± 1/2 smallest increment [ depending on thickness of note]. Elasped time 2 minutes or less

Sounds fine with the exception where dose the "find square root times 2" come from I always thought;

Circumference = π x Diameter

Did you thinking that you measured the area rather than the circumference in the first place?

Apart from that sounds good to me. Oh and by the way if it's a cheapie calculator that doesn't have π then 22 ÷ 7 is close enough for most engineering purposes.

Maybe overkill, but I usually use 3.14159. Most engineers and scientists I know, including myself, long ago memorized PI to at least 5 decimal places or more, much more accurate than 22/7 if you are using a calculator anyway.

Good Answer! (except for the oops) the accuracy will be better than .001 inches since you can divide half the smallest increment by PI as well!

"Measure w/thick steel rule.÷by pi find sq root x 2"

Why sq root?

'Pi multiplied by diameter' is the perimeter, which is what you have measured…

I will use the note to wrap round the cylinder and make a mark where it overlaps. I will then measure out that length with the steel rule. This measurement represents the circumference of the cylinder. To find the outside diameter, I will divide what'er I measured by 3.142 (pie).

I will use the same procedure to obtain the inside diameter. Then subtract inside diameter from outside diameter and divide result by 2 to obtain thickness.

Ok so my math sucks

Which is why I said eyeball it to make sure the answer, is in the right neighborhood.

Good point about the accuracy, being basically x3

Actually I would use the pi tape or caliper in my desk drawer!

Really the point of the Challenge was to be resourceful & use whats on hand?

1 Of my daily modes of operation

I would lay the pencil across the top of the cylinder, from end to end, then, using the steel ruler, look under the pencil to determine the gap.I would then subtract this amount x 2 from the total measurement from the desk to the top of the cylinder. This is presuming the flares are concentric with the center of the cylinder, and that the pencil is straight. Don't even need a calculator.

What about laying the rule onto the cylinder and gently rolling the cylinder over the desk until one full revolution is passed, then read off the lenght of the circumference from the rule.... I should have said mark the cylinder with the pencil... but not sure if you're allowed to do that?

If the circumference is small do multiple revolutions to give greater accuracy!!?

John.

Bonus Q: Like first reply, use paper from note on outside and measure to get Cout. Cout / pi = Dout. Next, fit paper inside and mark overlap, measure to get Cin. Cin / pi = Din. Wall thickness = (Dout - Din)/2. There are a number of points at which errors can be introduced in the process. I remember having to go through a "propagation of error" calculation exercise as part of a P-Chem experiment in college and I swore I would never go through that hell again! So no accuracy statement on this part!

This might be easier to explain with models, but here goes;

Step 1. Open desk drawer and lay cylinder, on side, against drawer bottom with END of cylinder towards drawer SIDE, and long portion of cylinder parallel to drawer FRONT.

Step 2. Hold one EDGE of boss' (observe that boss is spelled with a lower case "b", in that my keyboard can't find a lower case to assign the appropriate reverence that "it" deserves), note paper against the drawer side, with paper face parallel to drawer bottom, and slide to make contact with largest diameter of cylinder, which is also coincindental to mid point of cylinder's small diameter.

Step 3. Now, with scale, measure distance between drawer FACE and paper edge (for the largest diameter, say, DISTANCE LARGE,or, D(L)), and, also from paper edge to the narrowest diameter cylinder diameter, say, DISTANCE delta,or, D(d).

Step 4. Seems that cylinder's narrowest diameter D(N) is: D(N)= D(L)-2(D(d)).

Step 5. Take calculater (an HP with RNP?) and figure how much you're being overworked and underpaid in this job.

Step 6. Sanitize coffee mug and fill with fresh brew and enjoy with the PI that is left over that you didn't have to use.

Step 7. At this point, do we have to ponder the bonus point?

Is GOD an electrictian when HE said "LET THERE BE LIGHT"?

"What Man invents, Man can circumvent."

For simplicty, can't beat #8.

Okay, we have a number of solutions and all will work to a degree of satisfaction. However, I think Garthh has the highest probability to meet the challenge. That is, if we dust off his math a little. ;-)

#8 was interesting, but the issue is how do you accurately determine the distance between the pencil and the top of the curve of the central portion of the cylinder? The pencil width will overhang the top of the curve of the cylinder, so measuring the gap is tricky. Since you must multiply the gap by two you increase the error by two.

The advantage of measuring the circumference is that it improves your chances of accuracy by approximately 3.14159. Emmanuel did just this in post #6, too.

Paper thickness is typically between 0.004" to 0.005", so the slight amount of paper that overlaps itself when wrapping it around the circumference produces a nominal amount of error. You can mitigate that error if you pencil mark the point where the overlap occurs.

As it has been pointed out, the correct formula we want is Circumference = 2 * r * PI. Since 2r = the diameter, we can solve for diameter with:

d = Circumference / PI

The next question is how accurately can we measure the span of the unrolled paper? If our scale is graduated in increments of 0.050", the first thought would be a certainty level of 0.050". Actually, we can do much better than that, particularly because we have a magnifying glass. A good set of eyes should be able to split the distance between two 0.050" marks into quarters with an accuracy of +/- half that quarter distance. That works out to 12.5 thousands +/- 6.25 thousands! That isn't bad! Of course, a lot will depend on how accurately we mark with our pencil.

The challenge did not state how sharp the tip of the #2 pencil was, but did state that the eraser was chewed off. That means that the brass band that holds the eraser would be exposed. Using the back end of the pencil you should be able to mark the interface where the paper overlaps about the cylinder with pretty good precision. That band is about 0.015" to 0.020" thick, so let's use 0.020" (+/- 0.10") for our uncertainty when we make our measurement.

Next, unroll the paper on a flat surface and lay the scale so that the thin edge of the scale lies on the paper. One mistake many people do is to measure from the end of the scale. The end usually gets beat up and can not be depended on to be exactly the start of the scale, so slide the scale over to the 1" mark, make your best measurement, and subtract whatever you use as your starting dimension (i.e., 1 inch).

To improve your confidence it would be wise to repeat the above steps a number of times and take the average of the results.

Now remember that we divide by PI, or 3.14159, so our accuracy will improve by that ratio. So far our error budget is 0.00625 + 0.010" or a total of +/-0.01625". If we divide that by 3.14159 we get +/-0.00517"! Now that is the theoretical limit, and our actual results will probably vary. However, even if we are almost 3 times worse than that we are still at about +/-0.015".

To get the wall thickness we simply place the paper on the inside and repeat our process. Arguably, it will be more difficult to approach the same error budget we got with the outside measurement due to difficulties of working from the inside. Let's, for the sake of academics, say we are half as good as on the outside. It gives us a number we can pin down. Let's say that works out to +/-0.01034. Now the difference between the two diameters (inside and outside) divided by two should yield the wall thickness. The division should help us improve our accuracy again, but if we look at the best and worst cases for the two diameters we will get our wall thickness error.

The thinnest wall thickness case scenario is when the ID error is positive and the OD error is negative. The thickest scenario is when the OD is error is positive and the ID error is negative. The delta of these two errors, divided by two, should yield the error for the wall thickness. Since the error is symmetrical in the positive and negative directions, we only need to examine one case. The actual ID and OD are not relevant and we can for this experiment say that they are the same (wall thickness of zero). The error = (0.01034" + 0.00517") / 2 = 0.007755". That is less than 10 thousands of an inch for a theoretical limit.

All of these calculations are really a theoretical limit and your mileage will vary. We didn't talk about the accuracy of the scale, either. However, the lesson in this challenge is that we can improve on our measurements with the tactical use of division coupled with careful measurement techniques to get remarkable results with ordinary tools.

Remember, the boss said he wanted it FAST? # 8 has a very simple and elegant solution.It is easy to look under the pencil to find the gap measurement..as you rotate the spool one way, the gap will disappear.Rock the spool back and forth between visable and over the "Horizon".Record this number on the scale.Rotate it the other way till the numbers on the scale just disappear, and carefully rock it back and forth to get the exact measurement.You have just determined the height of the flare, as close as an eybell can get it.If you want to split hairs, use the magnifying glass to increase accuracy. Multiply this number by 2 and subtract from the total diameter to get the center of the spool diameter.If you wanted to REALLY split hairs, you could contstruct yourself a vernier scale using the paper and the ruler.

This will be your center of measurement, and the pencil should be at the exact top of the spool. With a fairly good set of eyes, the measurement error should be minimal.Remember, speed is of the essence.

# 8 has my vote.

"Pay no attention to the man behind the curtain" (Wizard of Oz)

While the task is pretty simple and quick, the accuracy is not going to be anywhere as close as you get with measuring the circumference and the time difference would not be that great.

The challenge was really to show how you can use simple techniques to get accurate results with tools at hand.

Try it both ways and be honest with your line-of sight below the pencil, then compare to the other methods.Time yourself both ways.Let me know how it comes out.Just curious.I have gotten very accurate results with the pencil method, (verified by caliper), and need no calculator or coffee mug.