6-Mar-2017: Calculate Propagated Uncertainty

When we measure a thing, it is hard to avoid making an error so we can put ± to show the uncertainty. However, how would the uncertainty changed after we calculate the uncertainty into some new numbers? The following lab shows the way of calculating propagated uncertainty. 

Purpose: Use the original uncertainty to calculate the new uncertainty. 


From the video we watched before the class, it gives us functions to calculate propagated uncertainty.

     dA = |∂A/∂L| • dL + |∂A/∂W| • dW          OR         df = √[(∂f • dx / ∂x)^2 + (∂f • dy / ∂y)^2]

By those functions, we can find out the propagated uncertainty.

Following is a problem on using functions to solve propagated uncertainty.

Measure the density of metal cylinders. Calculate the propagated error in each of your density measurement. 

Set up:

We used some measurement to measure weights, heights, and diameters of two metal cylinders.

Cylinder 1: m = 49.7 ± 0.1 g, h = 5.07 ± 0.01 cm, D = 1.25 ± 0.01 cm
Cylinder 2: m = 28.9 ± 0.1 g, h = 3.26 ± 0.01 cm, D = 1.28 ± 0.01 cm

Analyze:

Density = Mass / Volumn , Volumn = π • h • (D / 2)^2
∂V = √[(∂V • dh / ∂h)^2 + (∂V • dD / ∂D)^2] = √[(π • D^2 • dh / 4)^2 + (2π • h • D • dD / 4)^2]
∂ρ = √[(∂ρ • dm / ∂m)^2 + (∂ρ • dV / ∂V)^2] = √[(dm / v)^2 + (m • dV / V^2)^2]

Therefore, 
Cylinder 1: V = 6.22 ± 0.10 cm^3, ρ = 7.99 ± 0.13 g/cm^3
Cylinder 2: V = 4.19 ± 0.067 cm^3, ρ = 6.90 ± 0.11 g/cm^3

Conclusion:

Today, we learned two functions to calculate propagated uncertainty. 
     dA = |∂A/∂L| • dL + |∂A/∂W| • dW          OR         df = √[(∂f • dx / ∂x)^2 + (∂f • dy / ∂y)^2]
It allows us to calculate some new uncertainty with our calculation. When we measure things in real life, it helps us to describe objects more accurately.