I get two different answers. Which one is correct?

[carloz]

Homework Statement

M = (a-b)/2 + a

a = 15
b=5

What is the uncertainty in M if the uncertainty in a and b is ±0.7?

Homework Equations

for c = a + b
Error in c =√[(error in a)^2 + (error in b)^2]

The Attempt at a Solution

Error in M = √[0.7^2 * 3] = 1.2124

The problem I am having is that we learn that the above formula can only work when the errors are independent of one another. the error in a is obviously not independent of the error in a. so i think I'm wrong.

What do you think?

Thank you.

 

[rock.freak667]

If I remember correctly, the errors work like this:

s= a+b ⇒ Δs=Δa + Δb

s= a-b ⇒ Δs= Δa + Δb

s=ab ⇒ Δs/s = Δa/a + Δb/b

So you can apply the first two as needed.

 

[carloz]

yes. but that only works when the uncertainties in a and b are independent. however in my equation for M, a appears twice. since the error of a is not independent of a, how do i go about finding the uncertainty?

thanks.

 

[shfreeman]

Why not just simplify your problem

(a-b)/2+a

into

(3a-b)/2

and then use the rules for the error of 3a+b. You don't need to worry about the independency of error a to error a.
This way I get

(3 Δa + Δb)/2 = 1.4

Or using the other rule [ √(Δa² + Δb²) ]

( √(9*Δa² + Δb²) ) / 2 ≈ 1.1068

 

[asteriatic]

As a scientist, it is important to recognize and address any potential errors or discrepancies in our calculations. In this case, it appears that there may be a misunderstanding of the formula for calculating error in a sum or difference of two quantities. The formula you have provided is correct, but it assumes that the errors in a and b are independent of each other. In this case, the error in a is not independent of the error in b, as they both contribute to the error in M.

To properly calculate the uncertainty in M, we must use the formula:

Error in M = √[(error in a)^2 + (error in b)^2 + 2(r)(error in a)(error in b)]

Where r is the correlation coefficient between a and b. If a and b are perfectly correlated (r=1), then the error in M would be √[(error in a)^2 + (error in b)^2 + 2(error in a)(error in b)]. In this case, r is not specified, so we cannot accurately determine the uncertainty in M.

To address the issue of two different answers, it is possible that the first answer (1.2124) is incorrect due to the assumption of independent errors. The second answer would be correct if the correlation coefficient is 0, meaning that the errors in a and b are not related. It is important to carefully consider the assumptions and limitations of any formula used in calculations. In this case, it may be necessary to gather more information or clarify the problem statement to accurately determine the uncertainty in M.