A Mathematical Toolkit. Introduction: Chapter 2. Objectives

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A Mathematical Toolkit

Introduction: Chapter 2

1 About Science

When the ideas of science are expressed in mathematical terms, they are unambiguous.

The equations of science provide compact expressions of relationships between concepts.

The methods of mathematics and experimentation have led to enormous successes in science.

Mathematics—The Language of Science

Objectives

Define the SI standards of measurement.

Use commen metric prefixes.

Estimate measurements and solutions to problems.

Perform arithmetic operations using scientific notation.

SI units for Physics

The SI stands for "System International”. There are 3 fundamental SI units for LENGTH, MASS, and TIME.

They basically breakdown like this:

SI Quantity SI Unit

Length Meter

Mass Kilogram

Time Second

Of course there are many other units to consider. Many times, however, we express these units with prefixes attached to the front.

This will, of course, make the number either larger or smaller. The nice thing about the prefix is that you can write a couple of numbers down and have the unit signify something larger.

Example: 1 Kilometer – The unit itself denotes that the number is actually larger than "1" considering fundamental units. The fundamental unit would be 1000 meters

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Metric Measurement

• Most scientists use the metric system and SI units when collecting data and performing experiments

• Metric System - decimal system of measurement based on certain physical standards and scaled on multiples of 10

• (page 16)

Metric Prefixes

•

• These two measurements are different.These two measurements are different.

15 meters

15 meters 15 kilometers15 kilometers

•• How?How?

Metric Prefixes

•

• These two measurements are different.These two measurements are different.

15 meters

15 meters 15 15 kilokilometersmeters

•• How?How?

–

One has a

One has a prefix

prefix

Metric Prefixes

•

• You will need to convert between different metric units You will need to convert between different metric units like the following

like the following

15 meters

15 meters 15 kilo15 kilometersmeters

•

• How do you do this?How do you do this?

Metric Prefixes

•

• You will need to convert between different metric units You will need to convert between different metric units like the following

like the following

15 meters

15 meters 15 15 kilokilometersmeters

•

• How do you do this?How do you do this?

–

Know your prefixes and how they relate to the

base unit

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Metric Prefixes

•

• A prefix will make the unit bigger or smaller than the A prefix will make the unit bigger or smaller than the base unit

base unit

•

• We will use the base unit meter in this exampleWe will use the base unit meter in this example

•

• But remember:But remember:

•• Prefixes have the same number meaning when attached Prefixes have the same number meaning when attached to other units like seconds, liters, etc.

to other units like seconds, liters, etc.

Metric Prefixes

•

• Bigger Metric Prefixes Bigger Metric Prefixes

•• Prefix: deca-Prefix: deca-

•

• Symbol: daSymbol: da

•• Meaning: tenMeaning: ten

•

• 1 dam or decameter equals 10 meters1 dam or decameter equals 10 meters

Metric Prefixes

•• Bigger Metric Prefixes Bigger Metric Prefixes

•

• Prefix: hecto-Prefix: hecto-

•

• Symbol: hSymbol: h

•

• Meaning: one hundredMeaning: one hundred

•• 1 hm or hectometer equals 100 meters1 hm or hectometer equals 100 meters

Metric Prefixes

•• Bigger Metric Prefixes Bigger Metric Prefixes

•

• Prefix: kilo-Prefix: kilo-

•

• Symbol: kSymbol: k

•

• Meaning: one thousandMeaning: one thousand

•• 1 km or kilometer equals 1000 meters1 km or kilometer equals 1000 meters

Metric Prefixes

•

• Bigger Metric Prefixes Bigger Metric Prefixes

•

• Prefix: mega-Prefix: mega-

•• Symbol: M Symbol: M capital M capital M

•

• Meaning: one millionMeaning: one million

•• 1 Mm or Megameter equals 1000000 meters1 Mm or Megameter equals 1000000 meters

Metric Prefixes

•

• Smaller Metric Prefixes Smaller Metric Prefixes

•

• Prefix: deci-Prefix: deci-

•• Symbol: d Symbol: d

•

• Meaning: one tenthMeaning: one tenth

•

• 10 dm equals 1 m10 dm equals 1 m

1 dm 1 dm

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Metric Prefixes

•

•• Prefix: centi-Prefix: centi-

•

• Symbol: c Symbol: c

•• Meaning: one hundredthMeaning: one hundredth

•

• 100 cm equals 1 m100 cm equals 1 m

1 cm 1 cm

Metric Prefixes

•

•• Prefix: milli-Prefix: milli-

•

• Symbol: m Symbol: m

•• Meaning: one thousandthMeaning: one thousandth

•

• 1000 mm equals 1 m1000 mm equals 1 m

1 mm 1 mm

Metric Prefixes

•• Smaller Metric Prefixes Smaller Metric Prefixes

•

• Prefix: micro-Prefix: micro-

•

• Symbol: µSymbol: µ

•

• Meaning: one millionthMeaning: one millionth

•• 1000000 µ1000000 µm equals 1 mm equals 1 m

•• Too small to read on a meter stickToo small to read on a meter stick

Metric Prefixes

SI PREFIX

Mega- kilo- hecto- deca-

None deci- centi- milli- micro-

Unit Symbol

M k h da ---- d c m µ

Meaning in words

million thousand hundred ten

---- tenth hundredth thousandth millionth

Mathematical meaning

1,000,000 1,000 100 10 1 1/10 1/100 1/1000 1/1000000

Most commonly used prefixes in Physics

Prefix Factor Symbol

Mega- ( mostly used for radio station frequencies) x 10⁶ M

Kilo- ( used for just about anything, Europe uses the Kilometer instead of the mile on its roads)

x 10³ K

Centi- ( Used significantly to express small distances in optics. This is the unit MOST people in AP forget to convert)

x 10^-2 c

Milli- ( Used sometimes to express small distances)

x 10^-3 m

Micro- ( Used mostly in electronics to express the value of a charge or capacitor)

x 10^-6 µ

Nano ( Used to express the distance between wave crests when dealing with light and the electromagnetic spectrum)

x 10^-9 n

Tip: Use your constant sheet when you forget a prefix value

Conversions outside the metric system LENGTH

1 in = 2.54 cm 1 ft = 0.3048 m 1 mi = 5280 ft = 1.609 km 1 m = 3.281 ft

1 km = 0.6214 mi MASS 1 oz = 28.35 g 1 kg = 2.2 lb TIME

1 hr = 60 min = 3600 s

1 day = 24 h = 1.44 x 10³min = 8.64 x 10⁴ s 1 yr = 365 days = 3.156 x 10⁷ s

SPEED

1 mi/h = 1.609 km/h = 1.467 ft/s = 0.4470 m/s 1 km/h = 0.6214 mi/h = 0.277 m/s = 0.9113 ft/s

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Estimates

• The ability to make rough estimates without actually solving a problem is a skill you will find useful all your life.

• Rough estimates have to start with

reasonable lengths, times, weights, etc...

• Arrange these in order of largest to smallest measure of length

dm dam dm dam mm

mm mm hmhm

km cm km cm

Km hm dam m dm cm mm

• Arrange these in order of largest to smallest measure of length

dm dam dm dam mm

mm mm hmhm

km cm km cm

• Place a 1 in front of the unit that is bigger

• Next write how many of the smaller unit fit in the bigger unit

Scientific Notation

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Doing a conversion

• Convert 65 m to km 1.Turn this into a question

?km = 65m

Doing a conversion

• Convert 65 m to km 1. Turn this into a question

?km = 65m

2. Draw the conversion line

3. Write the unit we are converting out of on the opposite side of the line (so it will cancel out later) 4. Write what you are converting into on the opposite side

5. Place a 1 next to the larger unit

6. Place the correct conversion factor next to the other

m

km 1 1000

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Doing a conversion

?km = 65m

m km 1 1000

7. Cross out the units that have canceled out 8. Check to see if you are now at the desired unit 9. If yes.. Grab a calculator

If no.. You will need another conversion line

Doing a conversion

65

1 1000

Using your calculator to get a successful result

1.Open parenthesis, multiply everything in the numerator, close parenthesis

2.Hit divide

3.Open parenthesis, multiply everything in the numerator, close parenthesis

4.solve

( x ) ( )

0.065

Get some practice 1. 32 cm to m

?m = 32 cm

Get some practice 2. 14 minutes to seconds

?s = 14 min

Get some practice 3. 507 cL to L

?L = 507 cL

Get some practice 4. 0.057 m to km

?km = 0.057 m

or 5.7 x10 or 5.7 x10^-5^-5

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Get some practice 5. 13 cm

³

to mL

?mL = 13 cm

³

(1cm

³

= 1 mL)

6. 3.004 L to mL

?mL= 3.004L

7. 86 kg to g

?g = 86 kg

8. 1.2 x 10

³

mi to km

?km = 1.2 x 10

³

mi

(0.62mi = 1km)

9. 0.15ft to inches

?in = 0.15ft

(1ft = 12in)

Next level of difficulty

1. 56 km to mm

?mm= 56 km

Not there yet Not there yet……

need another conversion line need another conversion line

Now were there Now were there

Calculator time Calculator time

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2. 29 kg to cg

?cg = 29 kg

= =

2900000 cg 2900000 cg or 2.9 x 10 or 2.9 x 10

⁶⁶

cg cg

3. 0.546 km to µm

? µm = 0.546 km

= 546000000

= 546000000 µ µm m

or

5.46 x 10 5.46 x 10

⁸⁸

µ µm m

4. 69,000 dg to kg

?kg = 69,000 dg

= 6.9 kg

Next level of problem 5. 25 km/min to m/s

? m = 25 km s 1 min

2. The numerator is now what you want, time to do the same with the denominator

1. Do exactly what we did before to convert

3. To cancel out a denominator we need to start by placing the same unit up top

The trick here is ignoring the part of The trick here is ignoring the part of the unit you are not working with the unit you are not working with

= 416.67 m/s

6. 1.352 km/h to mm/s

? mm = 1.352 km s 1 hr

7. 20 km to ft

? ft = 20 km

(1 m = 3.281 ft)

= 65620 ft

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Dimensional Analysis

Suppose we want to convert 65 mph to ft/s or m/s.

Dimensional Analysis is simply a technique you can use to convert from one unit to another. The main thing you have to remember is that the GIVEN UNIT MUST CANCEL OUT.

hour miles

65 60sec

min 1 min 60 1hour•

mile ft 1

5280 =

×

⇒ ×

1 60 60 1

5280 1 1 65

s 95 ft

s

95

ft

ft meter 281 . 3

1 =

×

⇒ × 281 . 3 1

1

95 29m/s

• Good Accuracy - Poor Precision

x

x x x

x

x x x

Poor Accuracy - Good Precision

x x

x x x

Good Accuracy - Good Precision

4. Dimensional Analysis

• a. No number without a unit, UNLESS IT IS A RATIO

• b. Dimension - qualitative nature of a physical item

– (1) Ex - “length” - the separation between two points

– (2) can use brackets to denote dimensions

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