How should I calculate uncertainties?

[Darkmisc]

Homework Statement

I'm writing up a prac, which involves calculating the uncertainty for 1/(xy)^0.5.

There are two formulas that I think apply:

Homework Equations

For R = xy, u(R)/R = u(x)/x + u(y)/y

and

using differentials in calculus

dR = (dR/dx)dx + (dR/dy)dy [note that dR/dr and dR/dy are partials]

The Attempt at a Solution

I found the latter formula in a calculus textbook. It does not appear in the lab manual. However, I'm thrown by the fact that R depends on the square root of xy. I'm not sure if the first formula applies in this case.

Does anyone know the right approach?


Thanks

 

[anathema]

for your question! Calculating uncertainty for a function that involves a square root can be tricky, but it can be done using the first formula you mentioned. In this case, we can rewrite the function as R = (xy)^0.5 and use the following steps to calculate the uncertainty:

1. Take the natural logarithm of both sides of the equation: ln(R) = 0.5ln(xy)
2. Use the chain rule to find the uncertainty in ln(R): u(ln(R)) = 0.5[(u(x)/x)+(u(y)/y)]
3. Convert back to the original function by raising e to the power of both sides: R = e^(ln(R))
4. Use the uncertainty in ln(R) to find the uncertainty in R: u(R)/R = u(ln(R))/ln(R)
5. Substitute in the expression for u(ln(R)) from step 2: u(R)/R = 0.5[(u(x)/x)+(u(y)/y)]
6. Finally, rearrange the equation to solve for u(R): u(R) = R[0.5(u(x)/x)+0.5(u(y)/y)]

Using this approach, you can calculate the uncertainty for 1/(xy)^0.5 by plugging in the values for R, x, and y. I hope this helps and good luck with your prac!