I Multiplying Uncertainties in Different Units

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Multiplying quantities with different units while including uncertainties involves using the standard propagation of uncertainty formula. In the example of calculating impulse (I = Ft), the uncertainties in force (F) and time (t) can be combined using their partial derivatives. If the uncertainties are uncorrelated, the covariance term drops out, simplifying the calculation. The discussion emphasizes that when plugging in the values, the units will naturally align, confirming the correctness of the approach. Understanding how to compute partial differentials is essential for accurate uncertainty propagation in such calculations.
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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
 
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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.
 
Dale said:
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?
 
e2m2a said:
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.
 
e2m2a said:
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.
 
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