in this video we're given measurements of the diameter and the height of an aluminum cylinder
we're told that we took these measurements with a set of calipers each of these measurements is
given a plus or minus uncertainty on it of .002 centimeters which is the actual uncertainty on
the calipers i have in the lab and what we're paying close attention to in this problem is
using the propagation of errors formulas to properly get the uncertainty in the volume of
this sample so we begin in part a by finding the volume of the sample in cubic centimeters and the
volume of a cylinder is just the area of the base multiplied by the height and one important point
in this problem is that to deal with uncertainty properly we need to state the volume formula in
terms of those things that were directly measured so instead of writing a radius there i'm going to
write diameter divided by 2. and when i square the diameter over 2 i get a factor of 1 4 and
i'll just pull that out in front so we're using the formula one quarter pi diameter squared times
the height so we just have to plug the numbers in now when it's one quarter pi the diameter is 3.226
the height 7.500 and i'm going to keep a lot of extra precision until we get to part b our volume
comes out to 61.303 cubic centimeters in part b we're asked to use the propagation of errors
formula to get the uncertainty in the volume and then what isn't stated in the instructions here
is that when we get that uncertainty our convention is to round the uncertainty to one
significant digit and then round off the original volume measurement to that same place value
so the volume is a product of a power of one measured quantity that's the diameter and then
another major quantity that's the height so the propagation of errors formula that we're going to
use looks like this i say the relative uncertainty in v in other words delta v over v is the square
root of 2 times delta d over d and the 2 is there because it was squared in the volume formula
and then i have to square that term plus a delta h over h squared inside that square
root so we plug in the numbers our uncertainty in the diameter was .002 the diameter itself
3.226 the uncertainty in the height .002 the height itself 7.500 and i get .00127
now this is the relative uncertainty in the volume in other words you think of it as a
percent uncertainty so it's about a one tenth of one percent to get the actual uncertainty
in the volume we need to multiply both sides by that v that's in the denominator so delta v
is point zero zero one two seven multiplied by my volume itself which was 61.303 cubic centimeters
and this gives me an uncertainty of .078 cubic centimeters as promised i'm going
to round this to one significant digit so my uncertainty is .08
cubic centimeters and then i'm going to round my original volume to that same decimal place
addressed by the uncertainty in other words the hundreds place because there's no reason to
include extra digits in my volume measurement if i have uncertainty in the hundreds place
the thousandths place becomes irrelevant so we state our final volume like this v is equal to
point 61.3 zero plus or minus point zero eight cubic centimeters if you enjoyed this video or
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