SI Units and Uncertainties

SI Units and Uncertainties

Unit 1: Measurements


SI Unit (Le Système International d’Unités)

Fundamental units meter (m) kilogram (kg) second (s) ampere (A) Kelvin (K) mole (mol) candela (cd)


Derived Units Any unit made of 2 or more

fundamental units m s-1

m s-2

Newton (N) = kg m s-2

Joule (J) = kg m2 s-2

Watt (W) = kg m2 s-3

Coulomb (C) = A s

Estimation with SI Units

Fundamental Units Mass: 1 kg – 2.2lbs / 1 L of H2O /

An avg. person is 50 kg Length: 1 m - Distance between one’s

hands with outstretched arms Time: 1 s - Duration of resting heartbeat

Derived Units Force: 1 N- weight of an apple Energy: 1 J- Work lifting an apple off of

the ground

Scientific Notation and Prefixes

SI prefixes Table

1 Gm = 1,000,000,000 m = 1,000,000 km1 GM = 1 x 109 m = 1 x 106 km

0.0000000001 s = 1 ?s = ? ms

Uncertainties & Errors

A. Random Errors1. Readability of an instrument2. A less than perfect observer3. Effects of a change in the

surroundings

Can be reduced by repeated readings

B. Systematic Errors1. A wrongly calibrated instrument2. An observer is less than perfect

for every measurement in the same way

Cannot be reduced by repeated readings

Uncertainties & Errors (cont.)

•An experiment is accurate if……it has a small systematic error

it has a small random error

x

x

x

x

Systematic error

Random errors

Perfect

•An experiment is precise if……

Uncertainties & Errors (cont.)

Accuracy and Precision:

Precise but not accurate

Accurate but not precise

Precise and accurate!

Precision– uniformityAccuracy- conformity

to a standard

Determining the Range of Uncertainty

1) Analogue scales (rulers,thermometers meters with needles)

±half of the smallest division

2) Digital scales

±the smallest division on the readout

If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g

10

40

30

20

50

Since the smallest division on the cylinder is 10 ml, the reading would be 32 ± 5 ml

Absolute Uncertainty- has units of the measurement

Range of Uncertainty (cont.)

3. Significant Figures

•The measurement is 14.742 g, the uncertainty of the measurement is 14.742 ± .001 g•The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml

If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure

Examples:

Range of Uncertainty (cont.)

4. From repeated measurements (an average)

Find the deviations between the average value and the largest and smallest values.

Example: A student times a cart going down a ramp 5 times, and gets these numbers: 2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 s Average: 1.98 s

The average is the best value and the largest deviation is taken as the uncertainty range:

Largest: 2.09 - 1.98 = 0.11 sSmallest: 1.98 - 1.89 = 0.09 s

1.98 ± 0.11 s

Mathematical Representation of Uncertainty

Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3

= g5.0 cm3

= 3.0 g cm-3

For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation.

Example 1:

Best value

mv

Density =

Mathematical Representation of Uncertainty

Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3

= g4.7 cm3

= 3.40 g cm-3

Example 1 (cont.):

Maximum value:

mv

Density =

Minimum value:

mv

Density = = g5.3 cm3

= 2.64 g cm-3

Mathematical Representation of Uncertainty (cont.)

•The uncertainty in the previous problem could have been written as a percentage

In this case, the density is 3.0 g cm-3 ± 13%

yy

= 3

X 100% = 13%

•The uncertainty range of our calculated value is the largest difference from the best value..

In this case, the density is 3.0 ± 0.4 g cm-3

Mathematical Representation of Uncertainty (cont.)

Example #2: What is the uncertainty of cos if = 60o ±5o?

•Best value of cos = cos 60o = 0.50•Max value of cos = cos 55o = 0.57•Min value of cos = cos 65o = 0.42

The largest deviation is taken as the uncertainty range:

In this case, it is 0.50 ± .08 OR 0.50 ± 16%

Deviates 0.07

Deviates 0.08

Mathematical Representation of Uncertainty: Shortcuts!

When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties.

Addition and Subtraction:

y = a + b Uncertainty of 2nd quantity

Uncertainty of 1st quantity

Total uncertainty

Mathematical Representation of Uncertainty: Shortcuts! (cont.)

•Determine the thickness of a pipe wall if the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm

Example for Addition and Subtraction:

Internal radius = 3.6 ± 0.1 cm

External radius = 4.0 ± 0.1 cm

Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm

Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm

Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%


The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity.

Multiplication and Division:

y = a + b + cy a b c Denominators

represent best values

Total percentage/ fractional uncertainty

Fractional Uncertainties of each quantity


Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3)

Example for Multiplication and Division:

y = a + by a b

= 1 + 0.3

15 5= 0.07 + 0.06 = .13 ( this means 13%)

13% of 3 g cm-3 is 0.4 g cm-3

3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%

The result of this calculation with uncertainty is:


Just multiply the exponent by the percentage (or fractional) uncertainty of the number.

For exponential calculations (x2, x3):

Cube- each side is 6.0 ± 0.1 cm Example:

Percent uncertainty

= 1.7%0.16

x 100 %=

Volume = (6 cm)3 = 216 cm3

Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3)

Therefore the volume is 216 ± 11 cm3

Problems:

1. If a cube is measured to be 4.0+_ 0.1 cm in length along each side.

Calculate the uncertainty in volume.

Answer: 64+_5 Cm

Problem ( IB 2010)

The length of each side of a sugar cube is measured as 10 mm with an uncertainty of +_2mm. Which of the following is the absolute uncertainty in the volume of the sugar cube?

a.+_6 mm c. +_400 mmb. +_8 mm d. +_600 mm

Problem:

3. The lengths and width of a rectangular plates are 50+_0.5 mm and 25+_0.5 mm. Calculate the best estimate of the percentage uncertainty in the calculated area.

a. +_0.02% c. +_3%b. +_1 % d. +_5%

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SI Units and Uncertainties