Understanding the Proper Uncertainty Formula to Use

[TRB8985]

Homework Statement

A rectangle is 7.60 +/- 0.01 cm long and 1.90 +/- cm wide. Find the area of the rectangle and the uncertainty in the area.

Relevant Equations

See post for provided formulas.

Good evening,

I'm running into a little confusion on the second part of this problem due to finding two different formulas for calculating the uncertainty in multiplied quantities.

The way that I was taught was something like this.

If $z = x \cdot y$, then: $$\dfrac{\delta z}{z} = \dfrac{\delta x}{x} + \dfrac{\delta y}{y}$$ Where $\delta x$, $\delta y$, and $\delta z$ represent the absolute uncertainties in $x$, $y$, and $z$ respectively.

With this approach, the uncertainty in the average area of this rectangle (14.4 cm) would be: $$\delta A = (14.4\: \text{cm}) \Big (\dfrac{0.01\: \text{cm}}{7.90\: \text{cm}} + \dfrac{0.01\:\text{cm}}{1.90\:\text{cm}} \Big) = 0.1\:\text{cm} $$ This matches what's listed in my textbook as the solution.

However, when looking a bit more into uncertainties online, I found this:

Statistics tells us that if the uncertainties are independent of one another, the uncertainty in a product is obtained by: $$\delta z = |z| \sqrt{\Big(\dfrac{\delta x}{x} \Big)^2 + \Big(\dfrac{\delta y}{y} \Big)^2 } $$ It's certainly true that the length and width of this rectangle are independent, and if I use the expression above, I still end up with the same answer. So then.. why the need for two formulas?

 

[Orodruin]

Indeed the formula to use depends on whether you consider the errors independent or dependent. In this case, the side lengths are presumably from two independent length measurements (although there could technically be a systematic part as well such as using the same ruler, which may have a slightly faulty grading etc).

The reason you are getting the same result is that the relative error on the short side is so much larger that it completely dominates the result.

 

[renormalize]

Homework Statement: A rectangle is 7.60 +/- 0.01 cm long and 1.90 +/- cm wide. Find the area of the rectangle and the uncertainty in the area.
Relevant Equations: See post for provided formulas.

Please check the units of area in your answer.

 

[TRB8985]

Please check the units of area in your answer.

Agreed! Looks like I missed the square there in $\delta A$ at the end.

Thank you both, have a great weekend.

 

[haruspex]

It depends what matters to you.

An engineer specifying manufacturing tolerances has to worry about worst case. If the bolt has diameter $r_b\pm \Delta r_b$ and the hole has diameter $r_h\pm \Delta r_h$ then it had better be that $r_b+ \Delta r_b<r_h- \Delta r_h$.
The first formula you quote takes that view. It assumes that the two errors reinforce.

The statistical version considers that the errors could mitigate each other. It interprets the input uncertainties as proportional to standard deviations and computes (to the same proportion) the standard deviation of the result. This is certainly more appropriate when many uncertainties are combined. If the masses of certain coins each have standard deviation 0.1g, independently, then the s.d. of the mass of 100 of them is 1g, as given by the second formula, not 10g as given by the first.