in this video we introduce formulas for the propagation of error which gives us the most
accurate way to compute the uncertainty in a quantity that's derived from multiple measurements
if you're already familiar with the significant digit rules for combining measurements those are
just a rough approximation of how to compute uncertainty while the propagation of error
formulas give us the real answer so just to define the notation real quick we're going to use x and y
for the measured quantities they have uncertainties of delta x and delta y
and then we use a z for the quantity that's derived by algebraically combining x and y
and then we use a delta z for the uncertainty in z and there's three different cases we're
going to look at the addition and subtraction case so if z is a sum or difference of x and y
then the uncertainty in z is given by something that looks a lot like the pythagorean theorem
you take the square root of the sum of the squares of the uncertainties in those individual measured
variables the multiplication and division case if z is proportional to a product or quotient of two
major quantities then we switch to looking at the relative uncertainty that's the uncertainty in z
divided by z in other words that's like a percent uncertainty and that again is equal to the square
root of the sum of the squares but this time it's the percent uncertainty in x and y that are being
squared inside that square root finally we look at the exponent formula if z is proportional to a
product of powers of x and y then we simply modify the multiplication and division formula by putting
those exponents inside the parentheses before we square those relative uncertainties of x and y
finally we're going to use a rounding convention to round our uncertainty to one significant digit
and then we're going to round the quantity z to that same decimal place as the uncertainty now
that's a lot to digest so we're going to clarify all this with three examples one for each formula
in our first example we're told that two masses are measured on scales that have
different uncertainties our first mass ends up being 2.34 plus or minus 0.01 kilograms
second is 0.75 plus or minus 0.02 kilograms and we want the sum of the masses and the correct
uncertainty on that sum so i'll go ahead and use this z for the sum and z is given by big m plus
little m and when i run the numbers that comes out to 3.09 kilograms so what about the uncertainty
in z that's given by the square root of the sum of the squares of the individual uncertainties
so the first one was 0.01 the second one was 0.02 when i run the numbers on this i get .022
for the uncertainty in the sum of the two masses and then we follow our rounding convention and
round that to one significant digit so it's .02 and finally we express our final answer
as 3.09 plus or minus .02 kilograms and just as a double check the significant digit in the
uncertainty should be the same place value as the last stated place value in the quantity itself
in the next example we're told that the speed of an object is measured
as 2.52 plus 0.05 meters per second and it moves in a straight line at that speed
in other words we have a constant velocity and that lasts for 0.58 plus or minus 0.02 seconds
we want the displacement of the object including the correct uncertainty so this time our
combined quantity is a product because to get the displacement i need to multiply speed times time
and we plug the numbers into that and we come out with a displacement of 1.4616 just keeping a lot
of extra digits for now and that's meters next we look at the uncertainty and this time because
it's a product we have to state everything in terms of relative uncertainty so i have delta
d over d equal to the square root of the sum of the squares of the relative uncertainties here
so i'm going to take delta v which was 0.05 and divide by v which was 2.52 we square that
and in the second term i have a 0.02 for the uncertainty in the time divided by the 0.58
for the value of the time when i run the numbers on this i get .0398
and then i have to do a little bit more in order to figure out the uncertainty on d
itself so in this formula i'm going to multiply both sides by d and that will isolate delta d
so i have delta d is equal to d multiplied by .0398 and this uncertainty comes out to .058
with units of meters and we're going to round that again to one significant digit so 0.06
then in the final statement of our answer d was 1.4616 but i'm going to round to the same decimal
place as the uncertainty so the hundreds place and i get 1.46 plus or minus that uncertainty delta d
so plus or minus 0.06 and don't forget the units in the final answer finally in our last example
we have a formula that involves exponents so i've measured the diameter of a cylinder
as d equals 1.55 plus or minus 0.02 inches and then the height is measured as 5.30 plus or
minus 0.02 inches i want the volume of this cylinder including the correct uncertainty
and there's kind of a geometry issue to work out before we get started on the uncertainty formulas
the volume of the cylinder is the area of the base multiplied by the height and if the base
is a circle as it is in this case that would be pi r squared and this brings up a tricky
issue our directly measured quantity was the diameter not the radius we really need to express
the volume in terms of the directly measured quantities so i have to plug in here that the
radius is half the diameter and when i square d over 2 it gives me a factor of 1 4. so i have 1 4
pi diameter squared times height now i can see that the volume is proportional to the diameter
squared and the heights to the first power and i can apply our formula so first we're going to find
the volume v and when i run the numbers on that it comes out to almost exactly 10 cubic inches
i'll just keep several decimal places for now then i apply my uncertainty formula and remember
because the diameter was squared a factor of 2 is going to come out in front of this term
and then i put in the uncertainty in d divided by d itself square the result
the second term is the uncertainty in h divided by h
all squared i'm going to add those up and square root the result
and when i run the numbers on that i get .0261
now again to solve for the actual uncertainty in v i need to multiply both sides of this
by v itself to isolate delta v so delta v is going to be v which was basically 10 10.001
times 0.0261 and this gives me 0.261 and again our convention here is to round uncertainties to
one significant digit so i'm going to call that 0.3 and then i'll round off my volume
to the same decimal point as the one addressed in the uncertainty in other words the tenths place
so v is going to be 10.0 plus or minus 0.3 don't forget the units that's cubic
inches in this case and we're done if you enjoyed this video or at least found it useful
check out another one by clicking one of the links on the left or click the zaxlab logo on
the right to explore dozens of physics and math playlists as always you can leave your questions
comments and requests in the comments section below and i'll get back to you within 24 hours
thanks for watching zach's lab and best of luck on your math and physics journey