Dimensional Analysis (a.k.a. unit conversions)

Dr. Chin Chu

Humans always want to quantify things.

Instruments were invented to accomplish

the task.

How to express the results from those

measurements?3.25meters

numerical value

unit

In science, numbers with no units do not mean anything!

Historically, many measuring systems were developed

based on differing units.

How to convert between different units?

Example: How many yards are there in 3.25 meters?

FACTOR-LABEL METHOD (a.k.a. DIMENSIONAL ANALYSIS)

A math protocol used to convert between different units.

Need: 1. Desired quantity.2. Given quantity.3. Conversion factors.

Conversion Factors: valid relationships or equities expressed as a fraction.

1 meter = 1.0936 yard

Example:

1 meter

1.0936 yard

1 meter 1.0936

yard Conversion Factors

Note:1. Basic conversion factors based on equivalency

relationships always equal to 1.2. They generally come in pairs with two units

trading places.

= 1 = 1

STEPS INVOLVED IN FACTOR-LABEL

METHOD:

1. Write down the given quantity /units.

2. Multiply given quantity/unit by 1’s.

3. Substitute 1’s with conversion factor(s)

that eliminate unwanted units and

achieve desired units.

4. Complete the math.

FACTOR LABEL EXAMPLE 1Q - How many yards are in 3.25 meters? (note: 1 m =

1.0936 yard)

First write down the given quantity/unit

3.25 meter

FACTOR LABEL EXAMPLE 1

Remember, anything multiply by 1 always stays

the same!

3.25 meter = 3.25 meter x1x1….

Actually, the 1’s could go on forever!

Next, multiply the given quantity/unit by 1’s.

FACTOR LABEL EXAMPLE 1

Remember, those conversion factors always equal to 1.

3.25 meter = 3.25 meter x1x1….

1 meter

1.0936 yard = 1

1 meter 1.0936

yard

= 1

So the 1’s in the above equation

could be substituted by those conversion factors. Here, only one substitution is

sufficient.

Which one?

FACTOR LABEL EXAMPLE 1

Choose the one that will cancel out the

meter.

3.25 meter = 3.25 meter x1

1 meter

1.0936 yard = 1

1 meter 1.0936

yard

= 1

FACTOR LABEL EXAMPLE 1

Finish the math.

3.25 meters = 1 meter

1.0936 yard

= 3.55 yard

Bravo! Mission accomplished.

Q - How many yards are in 3.25 meters? (note: 1 m = 1.0936 yard)

3.25 meter x

FACTOR LABEL EXAMPLE 1

If you are really good at this, an equivalent but simpler protocol

could be used.

THE MONKEY BAR:

3.25 meter = 3.25 meter

1 meter

1.0936 yard

= 3.55 yard

1 min.60 sec.

1 hour

60 min.

1 day24 hours

1 week7 days1 week

FACTOR LABEL EXAMPLE 2Q - How many seconds are there in a week? (note: 1

week = 7 days; 1 day = 24 hours; 1 hour = 60 minutes; 1 minute = 60 seconds)

A bit more complicated than Example 1. But the approach is the same.

1 week =

= 604,800 sec.

Often measurements return results of exceeding large or small values.

Examples:

2,345,600,000,000 dollars

0.000000000034 meter

How to make scientist’s life a bit easier?

Here comes the metric prefix system:

k m

Prefix(conversion factor)

unit

METRIC PREFIX SYSTEM

Prefix Name Prefix Symbol Conversion Factor Description yotta- Y 1024 1 septillion zepta- Z- 1021 1 sextillion exa- E- 1018 1 quintillion peta- P- 1015 1 quadrillion tera- T- 1012 1 trillion giga- G- 109 1 billion mega- M- 106 1 million kilo- k- 103 1 thousand

hector- h- 102 1 hundred deka- da- 10 1 ten deci- d- 10-1 1tenth centi- c- 10-2 1 hundredth milli- m- 10-3 1 thousandth

micro- 10-6 1 millionth nano- n- 10-9 1 billionth pico- p- 10-12 1 trillionth

femto- f- 10-15 1 quadrillionth atto- a- 10-18 1 quintillionth

zepto- z- 10-21 1 sextillionth yocto- y- 10-24 1 septillionth

Note: Prefixes in red color need to be mastered without aid of the table.

Metric Prefixes

What does the numerical values associated with metric prefixes mean?

It provides information regarding the numerical relationship

between unit with the prefix and unit without the prefix.

1 km = 103 m

Unit with prefix Unit with prefix

Factor

Conversion Factors:

1 km103 m 1 km

103 m or

10-9 m1 nm

1 Tm1012 m4 Tm

Factor-Label method could be readily applied to conversions between

metric units.

Common strategy:•Convert the given quantity/unit to unit without prefix.•Convert the unit without prefix to the desired unit.

4 Tm = ? nm

4 Tm = = 4 x 1021 nm