Dimensional Analysis (a.k.a. unit conversions) Dr. Chin Chu

In science, numbers with no units do not mean anything! Humans always want to quantify things. Instruments were invented to accomplish the task. How to express the results from those measurements? 3.25 meters numerical value unit In science, numbers with no units do not mean anything!

FACTOR-LABEL METHOD (a.k.a. DIMENSIONAL ANALYSIS) 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: Desired quantity. Given quantity. Conversion factors.

Conversion Factors: valid relationships or equities expressed as a fraction. Example: 1 meter = 1.0936 yard 1 meter 1.0936 yard 1 meter 1.0936 yard = 1 Conversion Factors Note: Basic conversion factors based on equivalency relationships always equal to 1. They generally come in pairs with two units trading places.

STEPS INVOLVED IN FACTOR-LABEL METHOD: Write down the given quantity /units. Multiply given quantity/unit by 1’s. Substitute 1’s with conversion factor(s) that eliminate unwanted units and achieve desired units. Complete the math.

First write down the given quantity/unit Factor label example 1 Q - How many yards are in 3.25 meters? (note: 1 m = 1.0936 yard) 3.25 meter First write down the given quantity/unit

Factor label example 1 3.25 meter = 3.25 meter x1x1…. Next, multiply the given quantity/unit by 1’s. Remember, anything multiply by 1 always stays the same! Actually, the 1’s could go on forever!

Remember, those conversion factors always equal to 1. Factor label example 1 3.25 meter = 3.25 meter x1x1…. 1 meter 1.0936 yard = 1 Remember, those conversion factors always equal to 1. So the 1’s in the above equation could be substituted by those conversion factors. Here, only one substitution is sufficient. Which one?

Choose the one that will cancel out the meter. Factor label example 1 3.25 meter = 3.25 meter x1 Choose the one that will cancel out the meter. 1 meter 1.0936 yard = 1 1 meter 1.0936 yard = 1

Bravo! Mission accomplished. Factor label example 1 Q - How many yards are in 3.25 meters? (note: 1 m = 1.0936 yard) 1 meter 1.0936 yard 3.25 meters = 3.25 meter x Finish the math. = 3.55 yard Bravo! Mission accomplished.

Factor label example 1 = 3.55 yard 3.25 meter 1.0936 yard 3.25 meter = THE MONKEY BAR: 3.25 meter 1 meter 1.0936 yard 3.25 meter = = 3.55 yard If you are really good at this, an equivalent but simpler protocol could be used.

A bit more complicated than Example 1. But the approach is the same. Factor label example 2 Q - 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) 1 week 1 week 7 days 1 day 24 hours 1 hour 60 min. 1 min. 60 sec. 1 week = = 604,800 sec. A bit more complicated than Example 1. But the approach is the same.

METRIC PREFIX SYSTEM Often measurements return results of exceeding large or small values. Examples: 0.000000000034 meter 2,345,600,000,000 dollars How to make scientist’s life a bit easier? Here comes the metric prefix system: k m Prefix (conversion factor) unit

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

What does the numerical values associated with metric prefixes mean? Unit with prefix Unit with prefix Conversion Factors: 1 km = 103 m 1 km 103 m or Factor It provides information regarding the numerical relationship between unit with the prefix and unit without the prefix. What does the numerical values associated with metric prefixes mean?

4 Tm = ? nm 4 Tm 1 Tm 1012 m 10-9 m 1 nm 4 Tm = = 4 x 1021 nm Common strategy: Convert the given quantity/unit to unit without prefix. Convert the unit without prefix to the desired unit. Factor-Label method could be readily applied to conversions between metric units.