Measurement

Why are precise measurements and calculations essential to a

study of physics?

Measurement & Precision

The precision of a measurement depends on the instrument used to measure it.

For example, how long is this block?

Measurement & Precision

Imagine you have a piece of string that is exactly 1 foot long.

Now imagine you were to use that string to measure the length of your pencil. How precise could you be about the length of the pencil?

Since the pencil is less than 1 foot, we must be dealing with a fraction of a foot. But what fraction can we reliably estimate as the length of the pencil?

Measurement & Precision

Suppose the pencil is slightly over half the 1 foot string. You guess, “Well it must be about 7 inches, so I’ll say 7/12 of a foot.”

Here’s the problem: If you convert 7/12 to a decimal, you get 0.583.

Can you reliably say, without a doubt, that the pencil is 0.583 and not 0.584 or 0.582?

You can’t. The string didn’t allow you to distinguish between those lengths… you didn’t have enough precision.

So, what can you estimate, reliably?

Measurement & Precision

Basically, you have one degree of freedom… one decimal place of freedom.

So, the only fractions you can use are tenths! You can only reliably estimate that the pencil is 0.6

ft long. It’s definitely more than 0.5 ft long and definitely less than 0.7 ft long.

Thus, precision determines the number of significant figures we use to report measurements.

In order to increase the precision of their measurements, physicists develop more-advanced instruments.

How big is the beetle?

Copyright © 1997-2005 by Fred Senese

Measure between the head and the tail!

Between 1.5 and 1.6 in

Measured length: 1.54 in

The 1 and 5 are known with certainty

The last digit (4) is estimated between the two nearest fine division marks.

How big is the penny?

Copyright © 1997-2005 by Fred Senese

Measure the diameter.

Between 1.9 and 2.0 cmEstimate the last digit.

What diameter do you measure?

How does that compare to your classmates?

Is any measurement EXACT?

What Length is Indicated by the Arrow?

Significant Figures

Indicate precision of a measured value 1100 vs. 1100.0 Which is more precise? How can you tell? How precise is each number? Determining significant figures can be tricky. There are some very basic rules you need to

know. Most importantly, you need to practice!

Counting Significant Figures

The Digits Digits That Count Example # of Sig Figs

Non-zero digits ALL          4.337 4

Leading zeros(zeros at the BEGINNING)

NONE          0.00065 2

Captive zeros(zeros BETWEEN non-zero digits)

ALL          1.000023 7

Trailing zeros (zeros at the END)

ONLY IF they follow asignificant figure AND

there is a decimalpoint in the number

89.00 but

8900

4 2

Leading, Captive AND Trailing Zeros

Combine therules above

0.003020 but

3020

4

3

Scientific Notation ALL         7.78 x 103 3

Calculating With Sig Figs

Type of Problem Example

MULTIPLICATION OR DIVISION:

Find the number that has the fewest sig figs. That's how many sig figs should be in your answer.

3.35 x 4.669 mL = 15.571115 mLrounded to 15.6 mL

3.35 has only 3 significant figures, so that's how many should be in the answer.  Round it off to 15.6 mL

ADDITION OR SUBTRACTION:

Find the number that has the fewest digits to the right of the decimal point. The answer must contain no more digits to the RIGHT of the decimal point than the number in the problem.

64.25 cm + 5.333 cm = 69.583 cm rounded to 69.58 cm

64.25 has only two digits to the right of the decimal, so that's how many should be to the right of the decimal in the answer. Drop the last digit so the answer is 69.58 cm.

Scientific Notation

Number expressed as: Product of a number between 1 and 10 AND a power of 10

5.63 x 104, meaning 5.63 x 10 x 10 x 10 x 10 or 5.63 x 10,000

ALWAYS has only ONE nonzero digit to the left of the decimal point

ONLY significant numbers are used in the first number First number can be positive or negative Power of 10 can be positive or negative

When to Use Scientific Notation

Astronomically Large Numbers mass of planets, distance between stars

Infinitesimally Small Numbers size of atoms, protons, electrons

A number with “ambiguous” zeros 59,000

HOW PRECISE IS IT?

Powers of 10

Positive Exponents

000,101010101010

100010101010

100101010

1010

4

3

2

1

Exponent of Zero Means “1” 100 = 1

Powers of 10

Negative Exponents

0001.010

001.010

01.010

1.010

000,101

101

101

101

1014

10001

101

101

1013

1001

101

1012

1011

Exponent of Zero Means “1” 100 = 1

Converting From Standard to Scientific Notation

Move decimal until it is behind the first sig fig Power of 10 is the # of spaces the decimal moved Decimal moves to the left, the exponent is positive Decimal moves to the right, the exponent is negative

428.5 4.285 x 102

(decimal moves 2 spots left)

0.0004285 4.285 x 10-4

(decimal moves 4 spots right)

Converting From Scientific to Standard Notation

Move decimal point # of spaces the decimal moves is the power of 10 If exponent is positive, move decimal to the right If exponent is negative, move decimal to the left

4.285 x 102 428.5(move decimal 2 spots right)

4.285 x 10-4 0.0004285(decimal moves 4 spots left)

Systems of Measurement

Why do we need a standardized system of measurement? Scientific community is global. An international “language” of measurement allows

scientists to share, interpret, and compare experimental findings with other scientists, regardless of nationality or language barriers.

By the 1700s, every country used its own system of weights and measures. England had three different systems just within its own borders!

Metric System & SI The first standardized system of measurement: the

“Metric” system Developed in France in 1791 Named based on French word for “measure” based on the decimal (powers of 10)

Systeme International d'Unites(International System of Units) Modernized version of the Metric System Abbreviated by the letters SI. Established in 1960, at the 11th General Conference on

Weights and Measures. Units, definitions, and symbols were revised and simplified.

Components of the SI System

In this course we will primarily use SI units. The SI system of measurement has 3 parts:

base units derived units prefixes

Unit: measure of the quantity that is defined to be exactly 1

Prefix: modifier that allows us to express multiples or fractions of a base unit

As we progress through the course, we will introduce different base units and derived units.

SI: Base Units

Physical Quantity Unit Name Symbol

length meter m

mass kilogram kg

time second s

electric current ampere A

temperature Kelvin K

amount of substance mole mol

luminous intensity candela cd

SI: Derived Units

Physical Quantity Unit Name Symbol

area square meter m2

volume cubic meter m3

speed meter persecond

m/s

accelerationmeter per

second squaredm/s2

weight, force newton N

pressure pascal Pa

energy, work joule J

Prefixes

Prefix Symbol Numerical MultiplierExponentialMultiplier

yotta Y 1,000,000,000,000,000,000,000,000 1024

zetta Z 1,000,000,000,000,000,000,000 1021

exa E 1,000,000,000,000,000,000 1018

peta P 1,000,000,000,000,000 1015

tera T 1,000,000,000,000 1012

giga G 1,000,000,000 109

mega M 1,000,000 106

kilo k 1,000 103

hecto h 100 102

deca da 10 101

no prefix means: 1 100

PrefixesPrefix Symbol Numerical Multiplier

ExponentialMultiplier

no prefix means: 1 100

deci d 0.1 10¯1

centi c 0.01 10¯2

milli m 0.001 10¯3

micro 0.000001 10¯6

nano n 0.000000001 10¯9

pico p 0.000000000001 10¯12

femto f 0.000000000000001 10¯15

atto a 0.000000000000000001 10¯18

zepto z 0.000000000000000000001 10¯21

yocto y 0.000000000000000000000001 10¯24

Unit Conversions

Method “Staircase” Factor-Label

Type Visual Mathematical

What to do…

Move decimal point the same number of places as steps between unit prefixes

Multiply measurement by conversion factor, a fraction that relates the original unit and the desired unit

When to use…

Converting between different prefixes between kilo and milli

Converting between SI and non-SI units

Converting between different prefixes beyond kilo and milli

“Staircase” Method

Draw and label this staircase every time you need to use this method, or until you can do the

conversions from memory

“Staircase” Method: Example

Problem: convert 6.5 kilometers to meters Start out on the “kilo” step. To get to the meter (basic unit) step, we need

to move three steps to the right. Move the decimal in 6.5 three steps to the

right Answer: 6500 m

“Staircase” Method: Example

Problem: convert 114.55 cm to km Start out on the “centi” step To get to the “kilo” step, move five steps to the

left Move the decimal in 114.55 five steps the left Answer: 0.0011455 km

Factor-Label Method

Multiply original measurement by conversion factor, a fraction that relates the original unit and the desired unit. Conversion factor is always equal to 1. Numerator and denominator should be equivalent

measurements. When measurement is multiplied by

conversion factor, original units should cancel

Factor-Label Method: Example

Convert 6.5 km to m First, we need to find a conversion factor that

relates km and m. We should know that 1 km and 1000 m are

equivalent (there are 1000 m in 1 km) We start with km, so km needs to cancel when we

multiply. So, km needs to be in the denominator

km 1

m 1000

Factor-Label Method: Example

Multiply original measurement by conversion factor and cancel units.

m 6500km 1

m 1000km 5.6

Factor-Label Method: Example

Convert 3.5 hours to seconds If we don’t know how many seconds are in an

hour, we’ll need more than one conversion factor in this problem

seconds 13000 :Answer

(2) figs sig ofnumber eappropriat toround

seconds 12600minute 1

seconds 60

hour 1

minutes 60hours 5.3