Propogation of Uncertainties

[ncm2]

1. The problem statement, all variables and given/known data
1. A cylindrical can with radius of 3.1 +/- 0.4 cm and height 4.5 +/- 0.5 cm. What is the uncertainty in volume in cm cubed?

2. If theta= 0.40 +/- 0.02 radians, what is the uncertainty of y = tan (theta)?


2. Relevant equations
Volume of cylinder = Pie * r^2 * h


3. The attempt at a solution
1. Uncertainty of Volume/Volume = Pie * square root of [(4*(.4/3.1)^2) + (.5/4.5)^2]
My answer is 186 which does not seem right.

2. Uncertainty of y = tan (.02/.4)*(tan .40)
My answer is 6.1E-6, which seems too small to be correct.

[LowlyPion]

1. The problem statement, all variables and given/known data
1. A cylindrical can with radius of 3.1 +/- 0.4 cm and height 4.5 +/- 0.5 cm. What is the uncertainty in volume in cm cubed?

2. If theta= 0.40 +/- 0.02 radians, what is the uncertainty of y = tan (theta)?

2. Relevant equations
Volume of cylinder = Pie * r^2 * h

3. The attempt at a solution
1. Uncertainty of Volume/Volume = Pie * square root of [(4*(.4/3.1)^2) + (.5/4.5)^2]
My answer is 186 which does not seem right.

2. Uncertainty of y = tan (.02/.4)*(tan .40)
My answer is 6.1E-6, which seems too small to be correct.

In 1) taking the RSS of the uncertainties is fine, except that I would rather think that
the uncertainty would be ((.4/3.1)² + (.4/3.1)² + (.5/4.5)²)½
(lose the pi)

For 2) I would rather think that your range of uncertainty would be expressed as Tan(.4) ± ½*(Tan(.42) - Tan(.38) )

[Delphi51]

1. Uncertainty of Volume/Volume = Pie * square root of [(4*(.4/3.1)^2) + (.5/4.5)^2]

I don't follow this at all. I'm stumped at the first number (4) which doesn't appear in the given information. Why the square root? Do you have some formula that you are following?

There are several ways of estimating the error in the answer. The easiest one, used when there is only multiplying and dividing in the formula, is to find the % error in each given quantity and then add them up. In \pi r^2h you would count the % error in r twice since you effectively have r x r, and then add the % error in h. About 55% or +/- 75.

For the tan question - not multiplying - you will have to use some other technique. A primitive but correct method is to simply find the answer at tan of .4, then find it again at the maximum of the range, that is tan of .4 + .02. The difference in the two answers is the +/- you are seeking.

[LowlyPion]

There are several ways of estimating the error in the answer. The easiest one, used when there is only multiplying and dividing in the formula, is to find the % error in each given quantity and then add them up.

Adding the relative uncertainties is also a method that yields a more conservative error estimation (larger). In this case since the dimensions - 2 of them anyway - are independently measured, I'd prefer the RSS method. If that is the material of the course the OP is studying then of course he should use whatever is the practice in his course for multiplying and dividing uncertainties.

[ncm2]

Thank you very much for the help. My only question, for question 1, why omit pie? The reason I kept it is because I viewed it as a constant, so I thought that constants carry over to the uncertainty value as well.

[LowlyPion]

Thank you very much for the help. My only question, for question 1, why omit pie? The reason I kept it is because I viewed it as a constant, so I thought that constants carry over to the uncertainty value as well.

With the multiplication and division rule for propagation you are adding the relative uncertainty already. What you get is a relative number i.e. a percentage. If you want to express the error as a ± absolute #, then you would use the % relative error off the nominal calculated value of the result. For your case pi will be accounted for in the calculated result, so you don't want to put it in the relative error too.

e.g if it came out as ± 7% of a volume that was say 24 you could express it as either 24 ± 7% or 24 ± 1.7 .

You may be thinking of when you have absolute uncertainties, and you multiply by a constant with no error like pi, then you would multiply the absolute error by the constant.