Phys211C1 p1

Physical Quantities and Measurement

What is Physics?

Natural Philosophy

science of matter and energy

fundamental principles of engineering and technology

an experimental science: theoryexperiment

simplified models

range of validity

size

spee

d

Classical Mechanics

Quantum Mechanics

Relativistic Mechanics

Quantum Field Theory

Phys211C1 p2

Quantifying predictions and observations

physical quantities: numbers used to describe physical phenomena

height, weight e.g.

may be defined operationally

standard units: International System (SI aka Metric)

defined units established in terms of a physical quantity

derived units established as algebraic combinations of other units

Quantity Unit Length Time Mass Temperature Electric Current

meter (m) second (s) kilogram (kg) kelvin (K) ampere (A)

Phys211C1 p3

Scientific Notation: powers of 105,820 = 5.82x103 = 5.82E3.000527 = 5.28x10-4 = 5.28E-4note: 103 = 1x103 =1E3 not 10E3!

Prefix Abbre-viation

Powerof Ten

femtopiconanomicromillicentikilomegagiga

fpnµmckMG

10-15

10-12

10-9

10-6

10-3

10-2

103

106

109

1/1,000,000,000,000,0001/1,000,000,000,0001/1,000,000,0001/1,000,0001/1,0001/1001,0001,000,0001,000,000,000

How big (in terms of everyday life/other things) is a

meter nanometer gram

centimeter kilometer kilogram

Common prefixes

Phys211C1 p4

Dimensional Analysis: consistency of units

Algebraic equations must always be dimensionally consistent.

You can’t add apples and oranges!

cminchcm

ftinches

ftft

inchcm

cminch

48.30540.212

11

1540.2540.21

converting units

treat units as algebraic quantities

multiplying or dividing a quantity by 1 does not affect its value

s5sm

2m10

time speed distance

vtd

Phys211C1 p5

Units Conversion Examples

Example 1-1 The world speed record, set in 1983 is 1019.5 km/hr. Express this speed in m/s

Example how man cubic inches are there in a 2 liter engine?

Some Useful Conversion factors:1 inch = 2.54 cm1 m = 3.28 ft1 mile = 5280 ft 1 mile = 1.61 km

Phys211C1 p6

Significant Figures and Uncertainty

Every measurement of a physical quantity involves some error

random error

averages out

small random error accurate measurement

systematic error

does not average out

small systematic error precise measurement

0

5

10

15

20

number 0 0 0 0 0 0 0 0 0 4 8 15 13 5 3 0 0

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55

0

5

10

15

20

number 0 0 0 0 0 0 2 6 10 18 11 1 0 0 0 0 0

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55

0

5

10

15

number 0 0 0 0 0 0 1 0 4 4 12 10 6 4 3 2 1

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55

less precise less accurate

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Indicating the accuracy of a number: x ± x or x± x

nominal value: the indicated result of the measurement

numerical uncertainty: how much the “actual value” might be expected to differ from the nominal value

sometimes called the numerical error

1 standard deviationA measured length of 20.3 cm ± .5 cm means that the actual length is expected to lie between 19.8 cm and 20.8 cm. It has a nominal value of x = 20.3 cm with an uncertainty of x .5 cm.

fractional uncertainty: the fraction of the nominal value corresponding to the numerical uncertainty

percentage uncertainty: the percentage of the nominal value corresponding to the numerical uncertainty

025.cm3.20

cm5. xx

%5.2cm3.20

%5.2%100cm3.20

cm5.%100

xx

Phys211C1 p8

Uncertainties in calculations

Adding and subtracting: add numerical uncertainty

Multiplying or Dividing: add fractional/percentage uncertainty

Powers are “multiple multiplications”

bac

bac

or

bac

%100%100%100

bb

aa

cc

bb

aa

cc

bac

or

bac

aa

Ncc

ac N

Phys211C1 p9

More complex algebraic expressions must be broken down operation by operation

a = 3.13±.05 b = 7.14 ±.01 c = 14.44 ±.2%

x

cbax

ccc

c

bababa

ba

baba

bb

aa

baba

cbacbax

)(

1001

%100

)()()(

)(

)(

)()(

)(

Phys211C1 p10

Significant Figures: common way of implicitly indicating uncertainty

number is only expressed using meaningful digits (sig. figs.)

last digit (the least significant digit = lsd) is uncertain3 one digit

3.0 two digits (two significant figures = 2 sig. figs.)

3.00 three digits,etc. (300 how many digits?)

Combining numbers with significant digits

Addition and Subtraction: least significant digit determined by decimal places (result is rounded)

.57 + .3 = .87 =.9 11.2 - 17.63 = 6.43 = 6.4

Multiplication and Division: number of significant figures is the number of sig. figs. of the factor with the fewest sig. figs.

1.3x7.24 = 9.412 = 9.4 17.5/.3794 = 46.12546 = 46.1

Integer factors and geometric factors (such as ) have infinite precision

x 3.762 = 44.4145803 = 44.4

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Estimates and Order of magnitude calculations

an order of magnitude is a (rounded) 1 sig fig calculation, whose answer is expressed as the nearest power of 10.

Estimates should be done “in your head”

check against calculator mistakes!

Additional Homework: with

a = 3.13±.05 b = 7.14 ±.01 c = 14.44 ±.2%

evaluate expressions (nominal value and uncertainty expressed as numerical uncertainty and percentage uncertainty)

222/3

2 //

baxax

bcaxcabx

bcaxabcx