dimensional analysis (aka factor-label)

Dimensional Analysis (aka Factor-Label)

This technique involves the use of conversion factors and writing all measurements with both numerical values and the unit of measurement

A conversion factor is where you have the same amount (entity) represented by two different units of measurement with their corresponding numerical values

Conversion Factors• Here are some examples• 1 foot = __ inches• 1 kilometer = ____ meters• 1 inch = 2.54 centimeters• 1 gallon = __ quarts• 1 acre = 4840 square yards• 1 day = ___ hours

121000

4

24

Conversion factors…cont.Conversion factors for 1 ft = 12 in

foot 1inches 12

or inches 12foot 1

There are almost an infinite number of conversion factors that include meters:

mm 1000m 1

, cm 100m 1

, km 1

m 1000

m 1yards 0.9144

, inches 39.37m 1

, feet 3.28m 1

Conversion Factors….cont.

• One member of a dinner party orders a 16 ounce steak and another orders a one pound steak- Compare the two steaks

• They are the same since 16 oz dry wt. = 1 pound


• In grade school we learned that 1 gallon contained 4 quarts or stating that relationship as an equality:

• 1 gallon = 4 quarts• Since 1 gallon and 4 quarts

represent the same amount, we have a Conversion Factor


• Start with 1 gallon = 4 quarts• Dividing each side by 1 gallon we get this equation• 1 gallon = 4 quarts 1 gallon 1 gallon Since 1 gallon divided by 1 gallon

equals 1

• Our equality becomes: 1 = 4 quarts 1 gallon


• Again start with 1 gallon = 4 quarts • But this time we’ll divide each side of

the equality by 4 quarts • The resulting equation is• 1 gallon = 4 quarts 4 quarts 4quarts

Conversion Factors…. Cont.

• The right side of our equation becomes one because 4 quarts divided by 4 quarts is 1

• 1 gallon = 14 quarts

• Rearranging this becomes 1 = 1 gallon 4 quarts


• A mid-presentation summary• We know that 1 gallon = 4 quarts• Using a little mathematical magic• 1 gallon = 1 and 4 quarts = 1 4 quarts 1 gallon

• Why is this an important concept?


• Now a little math review…………….• What is 5 x 1?• What is 5 x 2 ? 2• Both expressions give you the same

answer- why?• Because 2/2 equals 1 and therefore the

second equation is just like the first and we did not change the initial value of 5.

Putting It TogetherHere’s An Example

• How many quarts are in 15 gallons ?• Remember we do NOT want to

change the amount represented by 15 gallons, only the units in quarts

• So we’ll use the conversion factor between gallons and quarts; that is

1 gallon = 4 quarts

Our Example continued…….

• We set it up like this: 15 gallons x 4 quarts 1 gallon• Cancel units • Do the math to complete the

problem• 15 x 4 quarts = 60 quarts

Every measurement must have a unit.

60 quarts

What do I need to do?• From the problem determine the

following:–Known quantity (number and units)

which is called the Given

– Identify what the Desired units are

–Conversion factor(s) needed (both universal and question specific)

Factor label exampleQ - How many kilometers are in 47 miles?

(note: 1 km = 0.621 miles)

First write down the desired quantity

# km

Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles)

Next, equate desired quantity to the given

quantity

# km = 47 mi

Factor label example


Now we have to choose a conversion

factor

# km = 47 mi


Pick the one that will allow you to cancel

out miles


# km = 47 mi 1 km 0.621 mi

0.621 mi 1 km



Multiply given quantity by chosen conversion factor

# km = 47 mi 1 km 0.621 mi

0.621 mi 1 km



Cross out common factors

# km = 47 mi x 1 km 0.621 mi




# km = 47 x 1 km 0.621



Are the units now correct?

# km = 47 x 1 km 0.621



Yes. Both sides have km as units.

# km = 47 x 1 km 0.621



Yes. Both sides have km as units.

= 47 x 1 km 0.621

# km



Now finish the math.

# km = 47 x 1 km 0.621

= 75.7 km



The final answer is 76 km (correct sig fig)

# km = 47 x 1 km 0.621

= 75.7 km


SummaryThe previous problem was not that hard.

In other words, you probably could have done it faster using a different method.

However, for harder problems the factor label method is easiest.

Let’s answer the beginning questions

• The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s.

• Which one is faster?• Both must be in the same units, so we must

convert one. • Does it matter which one?• NO.

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Factor label exampleQuestion: 27 mph is equal to how many m/s?

factors needed: 1 mi = 1.609 km 1 hr = 60 min

1000 m = 1 km 1 min = 60 sec

First write down the desired quantity

# m/s

Next, equate desired quantity to the given quantity


# m/s = 27 mi/hr

Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km

1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

Now we have to choose conversion

factors


# m = 27 mis 1 hr

x


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

Pick the one that will allow you to cancel out miles

1.609 km 1 mi

1 mi 1.609 km


# m = 27 mi1 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

Multiply given quantity by chosen conversion factor

1.609 km 1 mi


# m = 27 mi1 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X


1.609 km 1 mi


# m = 27 mi1 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X

Are the units now correct?

1.609 km 1


# m = 271 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X

NO, both sides aren’t equal

1.609 km 1


# m = 271 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X

Must choose another factor

1000 m 1 kmX


1.609 1


# m = 271 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X 1000 m 1X

Do units match?NO, must choose another factor

1.609 1


# m = 271 hrs


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X 1000 m 1X X 1 hr

60 minCross out common

factorsDo units match

1.609 1


# m = 27 1s


1 hr = 60 min 1000 m = 1 km 1 min = 60 sec

X

Do units match?

1000 m 1X

NO, must choose another factor

X 1 60 min


NO – must choose another factor

Factor label exampleQ – 27 mph is equal to how many m/s? factors needed:

1 mi = 1.609 km 1 hr = 60 min

1000 m = 1 km 1 min = 60 sec


1.609 1

# m = 27 1s X 1000 m

1X X 1

60 min



1 min 60 sX



1 mi = 1.609 km 1 hr = 60 min

1000 m = 1 km 1 min = 60 sec


1.609 1

# m = 27 1s X 1000 m

1X X 1

60



1 60 sX

Do units match?


1 mi = 1.609 km 1 hr = 60 min

1000 m = 1 km 1 min = 60 sec


1.609 1

# m = 27 1s X 1000 m

1X X 1

60



1 60 sX

Do units match?YES !


1 mi = 1.609 km 1 hr = 60 min

1000 m = 1 km 1 min = 60 sec

1.609 1

# m = 27 1s X 1000 m

1X X 1

60 1

60 sX

Do the math= 12.0675 m/s

= 12 m/s (correct sig fig)

• The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s. Which one is faster? How much faster?

• Human: 27 mph = 12 m/s

• Fish: 31 m/s

• Which one is fastest?• How much faster?

31m/s – 12 m/s = 19 m/s

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Working with metric/SIquantity base unit• length meter

• mass gram

• volume liter

SI Base UnitsBase Quantity Name Symbol

Length meter m

Mass kilogram kg

Time seconds s

Electric current ampere A

Thermodynamic temperature

Kelvin K

Amount of substance

mole mol

Luminous intensity

candela cd

Working with metric/SIBase Unit

grammeterliter

m

Working with metric/SIWhen converting within the metric system it is

helpful to remember:

“1 always goes with the prefix”the value of the prefix goes with

base unit

Conversions with metric/SI

• Example: How many meters are in 12 km? # m = 12 km

1 kmx 1000 m = 1.2 x 104 mm m

Base Unitgrammeterliter

Use chart to get

conversion


• Example: How many cm are in 1.3 m? # cm = 1.3 m x 1 cm

0.01 m


Use chart to get

conversion


• Example: How many cm are in 1.3 m? # cm = 1.3 m x 1 cm

0.01 m = 1.3 x 102 cm


When given a problem with 2 “prefixes” always go from prefix to base/base to prefix.

Example: converting cm to km – convert cm (prefix) to meters (base), then meters (base) to km (prefix).These are often referred to as “2 step problems”


• Example: How many km are in 2.7 x 104mm?

• This would be considered a “2-step problem”.

• There are 2 prefixes – km to mm


• Example: How many km are in 2.7 x 104mm? # km = 2.7 x 104mm

1 mmx 0.001 m


Use chart to get

conversion



1 mmx 0.001 m x 1 km

1000 m


Use chart to get

conversion



1 mmx 0.001 m x 1 km

1000 m =

0.027 km

• Take time now to work on the practice problems

• Ask questions if you need help!!

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dimensional analysis (aka factor-label)

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