Propagation of uncertainty with some constants

[happyparticle]

TL;DR

How to calculate the propagation of uncertainty with some constants

Hi,

I have a value $100 \pm 0.1$ and a function $\frac{V}{E} = \frac{1}{\sqrt{1 + (\omega r c)^2}}$ and I would like to find the uncertainty.
Where r = 1000 and c = $5 \cdot 10^{-8}$ are constants.
However, I'm not sure to understand how.

Here's what I think and did.
Since I multiply the uncertainty by a constant.

$\sigma= (1000 \cdot 5 \cdot 10^{-8}) \cdot 0.1 =$

and then for the power
$\sigma= \frac{2 (5\cdot 10^{-6}) \cdot (\sqrt{\omega r c)}}{\omega r c}$

Where I'm using this formula $\frac{\sigma_f}{f} = \frac{n \sigma_a}{a}$

 

[ergospherical]

Say $Q \equiv V/E$, then $\sigma(Q) = (\partial Q/\partial \omega) \sigma(\omega) = -(\omega r^2 c^2 / [1+(\omega rc)^2]^{3/2}) \sigma(\omega)$

 

[Dale]

Summary:: How to calculate the propagation of uncertainty with some constants

Hi,

I have a value $100 \pm 0.1$ and a function $\frac{V}{E} = \frac{1}{\sqrt{1 + (\omega r c)^2}}$ and I would like to find the uncertainty.

What variable is the value that you have? And what uncertainty do you want to find?

 

[happyparticle]

What variable is the value that you have? And what uncertainty do you want to find?

The value is 100 and since the value has an uncertainty If I use this value in a formula I have to take into account the propagation of this uncertainty.

 

[Dale]

The value is 100

The value of what is 100? I see three possible variables that could be 100, but you just keep saying that “the value” is 100 without any hint which value you are talking about.

I am done here. You have wasted people’s time here twice. I could have already answered your question if you had bothered to be clear either the first time or when I asked for clarification. I am not going to waste my time any more.

 

[happyparticle]

100 is just an arbitrary variable, it could be 1000 it could be 10. I gave an example to show exactly what I meant. I could use any other function with 1 variable and some constant. It could be $\omega$, r or c. That is why I put those 3 together. However, since I wrote that r = 1000 and c = 5E-8, I don't know what you mean?

I don't understand why you tell me this is not clear and that I waste people's time. I thought it was obvious that I was talking about $\omega$, I mean I didn't even think about if it was clear or not, I just wrote as it, since r and c has their values, I really thought it was obvious, my bad if it wasn't.

Maybe I thought It was clear because I don't understand this concept really well, which is possible and that's why I'm asking this question.

 

[sophiecentaur]

If the OP had only made it clearer by explicitly choosing one of ω, r or c then there would be no problem here. It was probably 'too obvious' for him.
It does show, however, how the algebraic approach make things much clearer (albeit a bit less friendly, initially). Just expanding the equation with ω replaced by ω+δω would show what's going on. Ignore terms higher than δω² and you're there.

 

[ergospherical]

@EpselonZero I wrote down the answer in post #2, does it make sense?

 

[sophiecentaur]

@EpselonZero I wrote down the answer in post #2, does it make sense?

Yes - except that it's very minimalist for the OP. It's actually harder, conceptually than your two line answer implies. I suspect that answer just appeared as a bit of a blur and he passed over it to encounter getting his backside kicked about bad presentation .
@EpselonZero The OP needed help digging himself out of the quandary.
I'd suggest that
1. Numbers should be avoided; stick with all the symbols until the end.
2. Add the uncertainty δω to ω and insert (ω+δω) where you had ω. Expand the resulting expression and ignore terms with δω² and then reach for your calculator.
I know that's the basis of Differential Calculus but there's no harm in being explicit when explaining things to an elementary question.