# Propagation of uncertainties Formulas and graphs.

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Propagation of uncertainties Formulas and graphs

Volume of a cylinder 12 3456 D = (2.9 ± 0.16) cm D = 2.9 cm  D = 0.05 cm+ 0.01 cm+ 0.1 cm 12 3456 h = 1.5 cm  h = 0.05 cm+ 0.01 cm+ 0.05 cm h = (1.5 ± 0.11) cm

Volume of a cylinder D = (2.9 ± 0.16) cm h = (1.5 ± 0.11) cm Result will have 2 significant figures V = ¼  D 2 h V = ¼  (2.9 cm) 2 1.5cm V = 9.907789463 cm 3 V = 9.9 cm 3 How sure can we be about the result? Lowest end: D=2.74 cm, h= 1.39cm V = 8.2 cm 3 (-17%) Highest end: D=3.06 cm, h= 1.61cm V = 11.8 cm 3 (+19 %)

Using physical quantities with uncertainty in a formula leads to calculation results with an uncertainty. How much uncertainty? How does the formula influence this uncertainty? Is there a way to predict this? Volume of a cylinder

Uncertainties and functions V = ¼  D 2 h d V D V D+  DD-  D DD VV Uncertainty in volume arising from uncertainty in diameter:

Propagation of uncertainty V = ¼  D 2 h Uncertainty in V = contribution from  D + contribution from  h Every regular equation has an error equation. Every error equation has one term for each measured quantity.

Volume of a cylinder D = (2.9 ± 0.16) cm h = (1.5 ± 0.11) cm V = 9.9 cm 3 V = (9.9 ± 1.9) cm 3 Relative error:  V/V  100% = 1.9/9.9  100% = 19%