We need numbers in order to accurately take measurements

• When executing the scientific method we must perform experiments measurements data

• Express measurements using units

• International System of Units (SI) aka: Metric system (meters, grams, seconds)

• English units (feet, slugs, seconds)

Unfortunately American students must learn both systems!

• Units allow us to describe things numerically

• Measurement standards – a fixed and reproducible value for the purpose of taking accurate measurements

Measurements

How do we know the length of a meter, yard?

• Egyptians, cubit

• King Louis XIV, foot

• Distance from equator to north pole

1/10,000,000, meter, from Greek metron for “measure”

• Modern standard, distance light,travels

in 1/299,792,458 s Distance from Earth to nearest 1022 m large galaxyOne light year 1016 mRadius of Earth 106 mHeight of person 2 mDiameter of a period in a sentence 10-4 mDiameter of a hydrogen atom 10-10 mDiameter of a proton 10-15 m

The mass of things

• Mass is the amount of matter an object contains – different than, something's weight!

• Mass is measured in kilograms, grams, milligrams, etc…• How do we know the mass of a kilogram? – the kilogram is defined to

be the mass of a cylinder of platinum-iridium kept at the international Bureau of Weights and Measures

• Quart of milk ~ 1 kg Typical Masses

Milk Way Galaxy 4 10 41 kg

Earth 6 10 24 kg

Space Shuttle 2 10 6 kg

Elephant 5400 kg

Automobile 1200 kg

Human 70 kg

Baseball 0.15 kg

Honeybee 1.5 10 –4 kg

Bacteria 10 –15 kg

Hydrogen Atom 1.67 10 –27 kg

Electron 9.11 10 –31 kg

The time of things• Define time? Even though we all have a sense of what time is, it’s difficult to give a good definition for it.• In scientific work we need to know: When did it happen? How long did it take? • How do we know the time of a second?

- originally defined in terms of a fraction of the average day(1 second = 1/86,400 of an average day)

- currently defined in terms of the frequency of radiation emitted from a cesium atom (called an atomic clock)

Typical Times

Age of the universe 5 10 17 s

Age of the Earth 1.3 10 17 s

Existence of Humans 6 10 13 s

Human Lifetime 2 10 9 s

One Year 3 10 7 s

One Day 8.6 10 4 s

Time between heartbeats 0.8 s

Human reaction time 0.1 s

One cycle of an AM radio wave 10 –6 s

One cycle of a visible light wave 10 –15 s

Converting Units of Measurement – Dimensional Analysis

• It is often the case that we must convert from one set of units to another.• Suppose we want to convert 316 ft to its equivalent in meters

meter

cminch

cmmile

feetfootinches

kmmile

meterskm

1100

154.2

15280

112

162.0

10001

Example: How many kilometers is 50,000 inches?

kilometerskilometersxx

xx

milekilometer

feetmile

inchesfoot

inches

27.162.0528012

111000,50

62.01

52801

121

000,50

The order that you apply the conversions makes no difference in the end!

these cancel !

left with the units wewant !

DOWNSLOWYESmph

hourkilometermiles

meterskilometers

sm

!3.22

1sec3600

162.0

10001

/10

Converting Units of Measurement – Dimensional Analysis

meter

cminch

cmmile

feetfootinches

kmmile

meterskm

1100

154.2

15280

112

162.0

10001

If I run 10 m/s in a school zone posted 20 miles/hour, am I speeding? Here we must convert two things: meters to miles, and secondsto hours

Conversions are a breeze with the metric system because it is based on powersof 10!

Converting Units of Measurement – Dimensional Analysis

meter

cminch

cmmile

feetfootinches

kmmile

meterskm

1100

154.2

15280

112

162.0

10001

Converting higher order units

If I have a house with 2,000 ft2 how many m2 does this correspond to?

1 m = 3.28 ft

2

2

2 9185283

10002 m

ft

meterft .

.,

Notice these powers match!

Converting Units of Measurement – Dimensional Analysis

Prefix Power ExamplesKilo- 1000, 103 Kilometer, Kiloliter, Kilogram

Hecto- 100, 102 Hectometer,Hectoliter,Hectogram

Deca- 10, 101 Decameter,Decaliter,Decagram

m, l, gr 1, 100 meter,liter,gram

Deci- 0.1, 10-1 Decimeter,Deciliter,Decigram

Centi- 0.01, 10-2 Centimeter,Centiliter,Centigram

Milli- 0.001, 10-3 Millimeter,Milliliter,Milligram

What if I had 10 milliliters and needed to convert this to kiloliters?

kLkLL

kL

mL

LmL 510100001.0

1000

1

1000

110

There’s a cooler way to do it and it involves my friend Hector!

Prefix Power ExamplesKilo- 1000, 103 Kilometer, Kiloliter, Kilogram Kind

Hecto- 100, 102 Hectometer,Hectoliter,Hectogram Hector

Deca- 10, 101 Decameter,Decaliter,Decagram Decked

m, l, gr 1, 100 meter,liter,gram Mr.

Deci- 0.1, 10-1 Decimeter,Deciliter,Decigram Deci

Centi- 0.01, 10-2 Centimeter,Centiliter,Centigram Cinema

Milli- 0.001, 10-3 Millimeter,Milliliter,Milligram Monday

Converting Units of Measurement – Dimensional Analysis

Kind Hector Decked Mr. Deci at the Cinema on Monday.

K H D M D C M Each word represents one of the powers of ten in themetric system!!

Converting Units of Measurement – Dimensional Analysis

K H D M D C M

So let’s look at how this works using the example we just did.

What if I had 10 milliliters and needed to convert this to kiloliters?

K H D M D C M

10.0 mL = ?? kL

K H D M D C M

Notice that I had to move over 6 letters to get to the “K” (or Kilo). So this corresponds to the number (and direction) of spaces I have to move my decimal!

10.0 mL = 0.00001 kL

Let’s try another example!

Converting Units of Measurement – Dimensional Analysis

meter

cminch

cmmile

feetfootinches

kmmile

meterskm

1100

154.2

15280

112

162.0

10001

We can use converting units to solve some neat problems.

How about this. If I know that a stack of 1,000 - $1 bills is = 1 inch in height

Could I jump over $1,000,000?

Where would we start?

dollarsinch

dollars1000

1000,000,1

ftinches

ftdollarsinch

dollars 8312

11000

1000,000,1

If I could jump this high I would be in the NBA!!

???

7-inch and 14-inch pizzas ($4.95, $13.95)

How much bigger is the 14-inch and which is the better buy?

Area of a circle = r2

7-inch Area = 153.9 inch2 14-inch Area = 615.7 inch2

Best Buy? (Price per inch2)

7-inch $ 4.95/153.9 inch2 14-inch $ 13.95/615.7 inch2

0.03 $/inch2 0.02 $/inch2

Look this stuff is even useful for everyday life!!

CAN YOU BELIEVE IT!!

Dimensional Analysis• Any valid physical formula must be dimensionally

consistent – each term must have the same dimensions

From the table:

Distance = velocity × time

Velocity = acceleration × time

Energy = mass × (velocity)2