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AP Physics Chapter 1

Measurement

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AP Physics

Turn in Contract/Signature Lecture Q&A Website: http://www.mrlee.altervista.org

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Measurement and Units

Physics is based on measurement. International System of Units (SI unit)

– Fundamental (base)quantities: more intuitive

– Derived quantities: can be described using fundamental quantities.

length, time, mass …

Speed = length / time Volume = length3

Density = mass / volume = mass / length3

Two kinds of quantities:

– Created by French scientists in 1795.

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Units

Unit: a measure of the quantity that is defined to be exactly 1.0.

Fundamental (base) Unit: unit associated with a fundamental quantity

Derived (secondary) Unit: unit associated with a derived quantity– Combination of fundamental units

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Standard Units

Standard Unit: a unit recognized and accepted by all.

Quantity Unit Name Unit Symbol

Length Meter m

Time Second s

Mass kilogram kg

– Standard: a reference to which all other examples of the quantity are compared.

– Standard and non-standard are separate from fundamental and derived.

Some SI standard base units

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Length

Standard unit: meter (m) Standard meter bar: International Bureau of Weights

and Measures near Paris Secondary standards: duplicates In 1983:

The meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.

Other (nonstandard) units: cm, km, ft, mile, …

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Time

Standard unit: second (s) One second is the time taken by 9,192,631,770

vibrations of the light (of a specified wavelength) emitted by a cesium-133 atom.

Other nonstandard units: min, hr, day, …

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Mass

Standard unit: kilogram (kg) Standard kilogram cylinder: International

Bureau of Weights and Measures near Paris  Other nonstandard units: g, Lb, ounce, ton, ..

Atomic mass unit (amu, u)

1 u = 1.6605402 10-27 kg

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Changing Unit: Conversion Factors

Conversion factor: a ratio of units that is equal to one.

160

min1

s1

min1

60

s s60min1 and

So two conversion factors:

s60

min1min1

60sand

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A few equalities (conversion Factors) to remember

1 m = 100 cm 1 inch = 2.54 cm 1 mile = 1.6 km 1 hr = 60 min 1 min = 60 s 1 hr = 3600 s

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Question?

Two conversion factors from each identity, but which one to use?

Depends on the unit we want to cancel. – If the unit we want to cancel is on the top with the

numerator, then for the conversion factor we must put that unit at the bottom with the denominator.

– If the unit we want to cancel is at the bottom with the denominator, then for the conversion factor we must put that unit on the top with the numerator.

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Example: 5 min = ___ s

60

min

s

300s

5min 5min 1

min cannot be cancelled out. Not good conversion factor.

Good conversion factor.

Does not work!

5min 5min

5min min

60s

1min 60s

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Practice:

Convert 12.3 m to cm

10012.3 12.3 1230

1

cmm m cm

m

1 100m cm

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Chain-link Conversion

60min

hr

60

min

s

Convert: 2 hr = ____ s

2 2hr hr 7200s

1 60min

1min 60

hr

s

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Practice:

12 m = ___ inch

10012 12 472

1 2.54

cm inchm m inch

m cm

1 100

1 2.54

m cm

inch cm

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Still simple? How about…

2 mile/hr = __ m/s

Chain Conversion

2 2mile mile

hr hr 1600m

mile

1

3600

hr

s

0.89m

s

1 1600

1 3600

mile m

hr s

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More practice:

2.54cm

inch

2232.cm

5 inch2 = _____ cm2

2 2.545 5

cminch inch inch

inch

2.54cm

inch

2 232.258 32.cm cm

2 25 5inch inch

1 2.54inch cm

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When reading the scale,

Estimate to 1/10th of the smallest division

6 7 cm.5

6.3 cm

– Draw mental 1/10 divisions– However, if smallest division is already too small,

just estimate to closest smallest division.

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Uncertainty of Measurement

All measurements are subject to uncertainties.

Uncertainties in measurement cannot be avoided, although we can make it very small.

Uncertainties are not mistakes; mistakes can be avoided.

Uncertainty

– External influences: temperature, magnetic field– Parallax: the apparent shift in the position of an

object when viewed from various angles.

experimental error

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Precision

Precision: the degree of exactness to which a measurement can be reproduced.

The precision of an instrument is limited by the smallest division on the measurement scale.

– Uncertainty is one-tenth of the smallest division.– Last digit of measurement is uncertain, the

measurement can be anywhere within ± one increment of last digit. Meter stick: smallest division = 1 mm = 0.001 m uncertainty is 0.0001 m

1.2345m: 1.2344m -1.2346m3 digits after decimal pt4 digits after

decimal pt

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Uncertainty and Precision

What is the uncertainty of the meterstick?0.0001m

What is the precision of the meterstick?0.001m

How precise is the meterstick?0.001m

estimate

certain

certain

Sometimes, when not strictly:

precision = uncertainty

Both the uncertainty and precision of a meterstick is 0.0001m

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Uncertainty and Precision

What is the uncertainty and precision of 1.234?

Uncertainty = 0.001

Precision = 0.01 or 0.001 (loosely)

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More precise = smaller uncertainty

Which is more precise, 12.34 or 2.345?

12.34: uncertainty = 0.01

2.345: uncertainty = 0.001

So, 2.345 is more precise.

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Accuracy

Accuracy: how well the result agrees with an accepted or true value

Accuracy and Precision are two separate issues.

ExampleAccepted (true) value is 1.00 m. Measurement #1 is 1.01 m, and Measurement #2 is 1.200 m.

Which one is more accurate? #1, closer to true value.

Which one is more precise? #2, precise to 0.001m, compared to 0.01m of #1

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Significant Figures (Digits)

1. Nonzero digits are always significant.2. The final zero is significant when there is a decimal

point.3. Zeros between two other significant digits are always

significant.4. Zeros used solely for spacing the decimal point are not

significant.

Example: 1.002300

0.004005600 7 sig. fig’s

7 sig. fig’s 12300 3 sig. fig’s

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Practice:

How many significant figures are there ina) 123000

b) 1.23000

c) 0.001230

d) 0.0120020

e) 1.0

f) 0.10

3

64

6

2

2

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Operation with measurements

In general, no final result should be “more precise” than the original data from which it was derived.

Too vague.

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Addition and subtraction with Sig. Figs

The sum or difference of two measurements is only as precise as the least precise one.

Example:16.26 + 4.2 = 20.46

Which number is least precise? 4.2

Precise to how many digits after the decimal pt? 1 So the final answer should be rounded-off (up or down) to how many digits after the decimal pt? 1

=20.5

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Practice:

1) 23.109 + 2.13 = ____

2) 12.7 + 3.31 = ____

3) 12.7 + 3.35 = ____

4) 12. + 3.3= ____

1) 23.109 + 2.13 = 25.239 = 25.24

2) 12.7+3.31 = 16.01 = 16.0Must keep this 0.

3) 12.7+3.35 = 16.05 = 16.1

4) 12. + 3.3 = 15.3 = 15. Keep the decimal pt.

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Multiplication and Division with Sig. Figs

The number of significant digits in a product or quotient is the number in the measurement with the least number of significant digits

Example:2.33 5.5 = 12.815

Which number has the least number of sig. figs? 5.5

How many sig figs? 2 So the final answer should be rounded-off (up or down) to how many sig figs? 2

=13.

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Practice:

2.33/3.0 = ___

2.33 / 3.0 = 0.7766667 = 0.78

2 sig figs 2 sig figs

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What about exact numbers?

Exact numbers have infinite number of sig. figs.

If 2 is an exact number, then 2.33 / 2 = __

2.33 / 2 = 1.165 = 1.17

Note: 2.33 has the least number of sig. figs: 3

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Prefixes Used with SI Units

Prefix Symbol Fractions

nano n × 10-9

micro × 10-6

milli m × 10-3

centi c × 10-2

kilo k × 103

mega M × 106

giga G × 109

1 m = 1 × 10-6 m 1 mm = 1 × 10-3 m

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Dimensional Analysis

What is the dimension of K if ?21

2K mv

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2K mv

2K m v

[x] = dimension of quantity x

Ignore

mass 2

length

time

2

2

lengthmass

time

2

2

MLor

T

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l

r l

r

When angle in unit of radian

radian 180o

1' 60"

1 60 'o

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1 AU

1 pc1”

61 92.9 10AU miles

180

1 60 '

1' 60"

o

o

rad

1" radx

1

1

AU

pc

1 1AU x rad pc x pc

1 ly = distance traveled by light in one year

HW 57

speed time

186,000 1mile

yrs

186,000 smile

ys

Convert 1 syr y

Conversion factor to convert61 92.9 10AU miles ly