Multiplying Uncertainties in Different Units

[e2m2a]

TL;DR Summary

How do you multiply quantities with their uncertainties when the units are different?

I could not find any clear explanation on multiplying quantities with different units while including their uncertainties. For example, how would you compute the following product with their uncertainties? 3.4 Newtons +/- .12 Newtons x 1.7 seconds +/- .23 seconds

 

[Dale]

You treat it the same as any other uncertainty. You have the formula $f(F,t)=F \ t$ so according to the standard propagation of uncertainty $$\sigma^2_f=\left( \frac{\partial f}{\partial F} \right)^2 \sigma^2_F + \left( \frac{\partial f}{\partial t} \right)^2 \sigma^2_t + 2 \frac{\partial f}{\partial F} \frac{\partial f}{\partial t} \sigma_{Ft}$$

Notice that the units work out naturally.

 

[e2m2a]

You treat it the same as any other uncertainty. You have the formula $f(F,t)=F \ t$ so according to the standard propagation of uncertainty $$\sigma^2_f=\left( \frac{\partial f}{\partial F} \right)^2 \sigma^2_F + \left( \frac{\partial f}{\partial t} \right)^2 \sigma^2_t + 2 \frac{\partial f}{\partial F} \frac{\partial f}{\partial t} \sigma_{Ft}$$

Notice that the units work out naturally.

Sorry, I don't understand your reply. A little over my head. Could you give me a concrete example?

 

[Dale]

Sorry, I don't understand your reply. A little over my head. Could you give me a concrete example?

In your case, if the uncertainty in $F$ is un correlated with the uncertainty in $t$ then $\sigma_{Ft}=0$ so the last term drops out. Then $$\frac{\partial f}{\partial F}=t$$ and $$\frac{\partial f}{\partial t}=F$$ and you already know $F = 3.4$, $\sigma_F = 0.12$, $t = 1.7$, and $\sigma_t= 0.23$. So just plug in and calculate.

 

[Ibix]

Sorry, I don't understand your reply. A little over my head. Could you give me a concrete example?

You are trying to calculate a quantity called impulse, $I$, which satisfies $I=Ft$. The standard error in $I$ is what you are looking for, and is $\sigma_I$ (@Dale called this $\sigma_f$). This relates to the standard errors in $F$ and $t$, $\sigma_F$ and $\sigma_t$ respectively, and their covariance, $\sigma_{Ft}$, through the formula Dale gave.

You gave us $\sigma_F$ and $\sigma_t$. Do you know how to calculate the partial differentials? If not, it's really easy - $\frac{\partial I}{\partial F}$ is the derivative of $I$ with respect to $F$ when everything else is treated as a constant. Finally, do you think your measurement error in $F$ depends on your measurement error in $t$? If yes, you need to measure the covariance. If not (which I would think is the case) then $\sigma_{Ft}=0$ and you can ignore the last term.

Plug in the numbers - you'll find that each term has units of $(\mathrm{Ns})^2$, so the units work out.