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Propagation of uncertainties

Propagation of uncertainties. Formulas and graphs. 3. 3. 4. 4. 5. 5. 6. 6. 1. 1. 2. 2. Volume of a cylinder. D = 2.9 cm D = 0.05 cm+ 0.01 cm+ 0.1 cm. h = 1.5 cm h = 0.05 cm+ 0.01 cm+ 0.05 cm. D = (2.9 ± 0.16) cm. h = (1.5 ± 0.11) cm. Volume of a cylinder.

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Propagation of uncertainties

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  1. Propagation of uncertainties Formulas and graphs

  2. 3 3 4 4 5 5 6 6 1 1 2 2 Volume of a cylinder D = 2.9 cm D = 0.05 cm+ 0.01 cm+ 0.1 cm h = 1.5 cm h = 0.05 cm+ 0.01 cm+ 0.05 cm D = (2.9 ± 0.16) cm h = (1.5 ± 0.11) cm

  3. Volume of a cylinder D = (2.9 ± 0.16) cm h = (1.5 ± 0.11) cm Result will have 2 significant figures V = ¼ p D2 h V = ¼ p (2.9 cm)2 1.5cm V = 9.907789463 cm3 V = 9.9 cm3 How sure can we be about the result? Lowest end: D=2.74 cm, h= 1.39cm V = 8.2 cm3 (-17%) Highest end: D=3.06 cm, h= 1.61cm V = 11.8 cm3 (+19 %)

  4. Volume of a cylinder • 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?

  5. V d Uncertainties and functions V = ¼ p D2 h V V Uncertainty in volume arising from uncertainty in diameter: D-D D+D D D

  6. Propagation of uncertainty V = ¼ p D2 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.

  7. Volume of a cylinder D = (2.9 ± 0.16) cm h = (1.5 ± 0.11) cm V = 9.9 cm3 V = (9.9 ± 1.9) cm3 Relative error: V/V 100% = 1.9/9.9  100% = 19%