1.3. Units and Significant Figures

The Scientific Method starts with making observations = precise and accurate measurements

• 1.3.1. SI Units

• 1.3.2. Unit Prefixes

• 1.3.3. Significant Figures (Significant Digits)

• 1.3.4. Round Off Error

1.3.1. Units of Measurement

• In physics, the metric system is (almost) always used.

• Standard: SI Units (Système International)– MKS:

• Meter (m) – Length

• Kilogram (kg) – Mass

• Second (s) – Time

– CGS:• Centimeter (cm) – Length

• Gram (g) – Mass

• Second (s) – Time

The Primary Base Units

Property SI Unit Symbol

Length Meter m

Time Second s

Mass Kilogram kg

Electric charge Coulomb C

Temperature Kelvin ˚K or K

Almost everything is physics can be described by combinations of these 5 base units.

The magnitude of these units is arbitrary and historical. The important thing is that everyone uses consistent standards!

Standards of Calibration for Units

Length: the meter

Then: one ten-millionth of the distance from the North Pole to

the equator (1793)

Now: the distance traveled by light in a vacuum in

1/299,792,458 of a second

Mass: the kilogram

Then: the mass of one liter of water.

Now: the mass of a particular platinum-iridium cylinder kept at

the Intl. Bureau of Weights and Standards in Sèvres, France.

Time: the second

Then: 1 day = 24 hours, 1 hour = 60 min, 1 min = 60 s

Now: the time for a particular radio wave from a cesium-133

atom to complete 9,192,631,770 oscillation cycles.

Unit Conversions

Unit conversions are best done using fractions.

Q: How many centimeters is 1.75 inches?

1.75 in ×? cm

? in

1.75 in ×2.54 cm

1 in= 4.45 cm

Length in Physics

Exponential Notation

Because of the large range in magnitudes, exponential notation is commonly used.

Addition and Subtraction:

2.1 × 107 m+ 3.2 × 107 m = 5.3 × 107 m

To add or subtract, the exponents and units must be the same.

Example

2.1 × 105 cm + 4.2 × 103 in =

Convert to same units:

2.1 × 105 cm + 4.2 × 103 in ×2.54 cm

1 in=

2.1 × 105 cm + 10.668 × 103 cm =

Convert to same exponents:

2.1 × 105 cm + 0.10668 × 105 cm =

2.1 + 0.10668 × 105 cm =

2.2 × 105 cm

Exponential multiplication and division

2.0 × 107 m× 3.0 × 105 m = 6.0 × 1012 m2

Exponents are added, units are multiplied.

3.0 × 107 m

2.0 × 105 s= 1.5 × 102

m

s

When dividing, exponents are subtracted.

What if the units in the denominator were m? cm?

Mass in Physics

Mass vs. Weight

• Mass is a measure of the amount of matter (primarily protons and neutrons) in an object– It is an intrinsic, unchanging property

• Weight is a measure of the gravitational force acting on an object– It varies depending on the object’s location

So, a 1 kg object weighs more on Earth than on the Moon. In orbit, it is “weightless”.

Time in Physics

Unit Prefixes designate powers of 10

Example:Diameter of a red blood cell:8 × 10-6 m = 8 µm

Convention to use powers of 103:

Mass (≈weight) of a honeybee:1.5 × 10-4 kg = 1.5 × 10-1 g =

0.15 g or 150 mg

These prefixes can be used withNon-metric units:

1 GB = 109 bytes

a) Speed of light 3.0 × 108 m/s =0.30 Gm/s

b) Radius of Earth 6380 km =6.38 Mm

c) Stefan-Boltzmann constant 5.7 × 10-8 W m-2 K-4 = 57 nW m-2 K-4

Exercise: Apply Standard Unit Prefixes

Changing a prefix can be approached as a unit conversion:

Applying Standard Unit Prefixes

3.0 × 108m

s×

1 Gm

109 m

= 3.0 × 10−1Gm

s

= 0.30Gm

s

Unit Prefix Conversions

Q: How many cm2 is 1.0 m2?

1.0 m2 ×100 cm

1 m×100 cm

1 m

= 1.0 × 104 cm2

Not 100 cm2!

Relevant Prefixes

• Does it make sense to describe the length of this room in nm, m or km?

• The radius of the Earth?

• The size of a virus?

Using appropriate prefixes can be helpful.

Using non-SI Units is Sometimes Allowed

Sometimes exceptions are made when using non-SI units makes physical interpretation easier:

“The average human lifetime is 2.4 Gs.”

versus

“The average human lifetime is 75 years.”Life expectancy at birth for 1987, NC for Health Statistics

Accessible interpretation is important for validation, and to build an understanding of magnitudes, relative relationships and phenomenology.

Question

You measure a wooden board to be 1.3 m long and 14 cm across. What is the area of one side?

1.3 m × 14 cm = 18.2 m ∙ cm

= 18.2 m ∙ cm ∙1 m

100 cm

= 0.182 m2

= 0.18 m2

Practice Unit Conversions and Prefixes.

Measurement and Precision• Measurement tools (instruments) typically have

an operational range (maximum and minimum) and limited precision (degree of exactness)

• Example: Spring scales– To what precision can these measure?– General rule: round to the nearest marking shown;

do not interpolate between markings– However if the distance between markings is large,

you may be able to interpolate one more digit– Can these scales be used to estimate the weight of a

single sheet of paper? 5 million sheets?

• The precision of any measurement is limited, and is indicated by using “significant digits”

Significant Figures (Significant Digits)

Significant Figures: the number of digits in a quantity that are known with certainty (reliability). This is important to record!

e.g. you measure with a ruler 25.2 cm ± 0.05 cm

25.2 3 significant figures 0.0345, 568

12.009 5 significant figures 5437600, 23.000

4300 2 significant figures 3500, 6.2, 0.060

0.005 1 significant figure 600, 0.1, 0.009

Rules for Significant Figures

• All nonzero digits are significant.

• Zeroes between nonzero digits are significant

1.001 has 4 sig figs.

• Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point.

0.001 oC has 1 sig fig.

• Trailing zeros on whole numbers are not significant.

27000 has 2 sig figs.

• Trailing zeros to the right of a decimal point are significant.

1.0 has 2 sig figs.

0.060 has 2 sig figs.

Exercise: Significant Figures

How many significant figures?0.0305

3 significant figures5437600

5 significant figures0.070

2 significant figures5.0

2 significant figures600

1 significant figure5.01

3 significant figures0.9

1 significant figure

Mathematics of Significant Figures

• The number of significant figures after multiplication or division is the number of significant figures in the least known quantity:

23.41 × 4.1 = 95.981 = 96 (round, don’t truncate)

• The number of decimal places after addition and subtraction is equal to the smallest number of decimal places in any of the input values:

23.41 + 4.1 = 27.51 = 27.5

Mathematics of Significant Figures

• Note: calculators do not keep track of significant digits!

0.200 + 0.300 = 0.5

• Keep in mind if any if the numbers are exact!

23.41 × 4 (exact) = 93.64

Exercises

• 4.34 + 2.1 = 6.44 =

= 6.4

• 3.14159 / 2 = 1.570795 =

= 2

• 3.14159 / 2.0 = 1.570795 =

= 1.6

• 100 – 4.67 = 95.33 =

= 100

On Precise Measurements

“I often say that when you can measure something and express it in numbers, you know something about it. When you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the stage of science, whatever it may be.”

—Lord Kelvin

1824-1907

Significant Digits: Round-off Error

The last digit in a calculated number may vary depending on

how it is calculated, due to sequential rounding off of

insignificant digits.

Example: You buy two items for $2.21 and $1.35, each with

8% tax. What is the total cost?

Method 1:

$2.21 × 1.08 = $2.3868, rounds to $2.39

$1.35 × 1.08 = $1.458, rounds to $1.46

Sum: $2.39 + $1.46 = $3.85

Method 2:

$2.21 + $1.35 = $3.56

$3.56 × 1.08 = $3.8448, rounds to $3.84

Method 2 is more accurate: round off insignificant digits only at

the very end of the calculation.