dimensional analysis powerpoint

8/11/2019 Dimensional Analysis Powerpoint

http://slidepdf.com/reader/full/dimensional-analysis-powerpoint 1/32

One of these things is Not

like the other…

This guide will explain briefly theconcept of units, and the use of a

simple technique with a fancy name—

"dimensional analysis”.



Whatever You Measure,

You Have to Use Units

• Suppose the label on a ball of stringindicates that the length of the string is

150. • Is the length 150 feet, 150 m, or 150 cm?

• For a measurement to make sense, it

must include a number and a unit.

• A measurement is a way to describe theworld using numbers. We usemeasurements to answer questions like,

how much? How long? How far ?



Units and Standards

• Now suppose you and a friend want tomake some measurements to find outwhether a desk will fit through a doorway.

• You have no ruler, so you decide to useyour hands as measuring tools.

150 feetNumber Unit

Rule: No naked numbers. They must have units



• Even though youboth used hands

to measure, youdidn’t check tosee whether yourhands were the

same width asyour friend’s.

Units and Standards



Units and Standards

• In other words, youdidn’t use ameasurement standard,

so you can’t comparethe measurements.

• Hands are aconvenient measuringtool, but using themcan lead tomisunderstanding.



Units and Standards

• So in order to avoid confusion we usemeasurement standards.

• A standard is an exact quantity thatpeople agree to use to comparemeasurements.



Units and Standards

• In the United States, we commonly useunits such as inches, feet, yards, miles,gallons, and pounds. This is known as

the English system of measurement.

• Most other nations and the scientificcommunity use the metric system

- a system of measurement based on multiples of ten.



International System of Units

• In 1960, an improvement was made tothe metric system. This improvement isknown as the International System ofUnits.

• This system is abbreviated SI from the

French Le Systeme Internationaled’Unites.



• The standard kilogram is kept in Sèvres,France.

• All kilograms used throughout the worldmust be exactly the same as the kilogramkept in France because it is the standard.





• Each type of SImeasurementhas a base unit.

• The meter is thebase unit of

length.




• Every type ofquantity measuredin SI has a symbol

for that unit.

• All other SI units are

obtained from theseseven units.



Review

• When we measure something, wealways specify what units we aremeasuring in.

• All kinds of units are possible, but inscience we use the SI system.

• Problem! What if I measure somethingin inches but I am supposed to giveyou the answer using SI units?



Sometimes You Have to Convert

Between Different Units

• How many seconds are in a day?

• How many inches are in a centimeter?

• If you are going 50 miles per hour,how many meters per second are youtraveling?

• To answer these questions you needto change (convert) from one unit toanother.



How do you change units?

• So what is dimensional analysis?

The converting from one unit system to another.

• Whenever you have to convert a physicalmeasurement from one dimensional unit to

another, dimens ional analysis is themethod used.

If this is all that it is, why make such a fuss about it? Verysimple. Wrong units lead to wrong answers. Scientists have

thus evolved an entire system of unit conversion.

(It is also known as the unit-factor method or the factor-label method)



Dimensional Analysis

• In order to perform any conversion, you need a convers ion factor .

• How does dimensional analysis work?

• It will involve some easy math (Multiplication & Division)

• Conversion factors are made from any two termsthat describe the same or equivalent “amounts”

of what we are interested in. For example, we know that:

1 inch = 2.54 centimeters1 dozen = 12



Conversion Factors

• In mathematics, the expression to the left of theequal sign is equal to the expression to the right.They are equal expressions.

• So, conversion factors are nothing more thanequalities or ratios that equal to each other. In“math-talk” they are equal to one.

or

• For Example

12 inches = 1 footWritten as an “equality” or “ratio” it looks like

= 1 = 1



Conversion Factors

or

• Conversion Factors look a lot like fractions, butthey are not!

Hey!These

look like

fractions!

• The critical thing to note is that the units

behave like numbers do when you multiply

fractions. That is, the inches (or foot) on top andthe inches (or foot) on the bottom can cancelout. Just like in algebra, Yippee!!



Example Problem #1

• How many feet are in 60 inches?Solve using dimensional analysis.

• All dimensional analysis problems are setup the same way. They follow this samepattern:

What units you have x What units you want = What units you wantWhat units you have

The number & unitsyou start with The conversion factor

(The equality that looks like a fraction)

The units youwant to end with



• Remember

12 inches = 1 footWritten as an “equality” or “ratio” it looks like

60 inches

Example Problem #1 (cont)

• You need a conversion factor. Somethingthat will change inches into feet.

What units you have x What units you want = What units you want

What units you have

x = 5 feet

(Mathematically all you do is: 60 x 1 12 = 5)



• This format is more visually integrated,

more bridge like, and is more appropriatefor working with factors. In this format, thehorizontal bar means “divide,” and thevertical bars mean “multiply”.


• The previous problem can also be writtento look like this:

• 60 inches 1 foot = 5 feet

12 inches




• The hardest part about dimensionalanalysis is knowing which conversionfactors to use.

• Some are obvious, like 12 inches = 1 foot,while others are not. Like how many feetare in a mile.



Example Problem #2

• You need to put gas in the car. Let'sassume that gasoline costs $3.35 pergallon and you've got a twenty dollar bill.

How many gallons of gas can you get withthat twenty? Try it!

• $ 20.00 1 gallon = 5.97 gallons

$ 3.35

(Mathematically all you do is: 20 x 1 3.35 = 5.97)



Example Problem #3

• What if you had wanted to know not how manygallons you could get, but how many miles youcould drive assuming your car gets 24 miles agallon? Let's try building from the previousproblem. You know you have 5.97 gallons in thetank. Try it!

• 5.97 gallons 24 miles = 143.28 miles

1 gallon

(Mathematically all you do is: 5.97 x 24 1 = 143.28)



Example Problem #3

• There's another way to do the previoustwo problems. Instead of chopping it upinto separate pieces, build it as one

problem. Not all problems lendthemselves to working them this way butmany of them do. It's a nice, elegant wayto minimize the number of calculations

you have to do. Let's reintroduce theproblem.



• $ 20.00 1 gallon 24 miles = 143.28 miles

$ 3.35 1 gallon


• You have a twenty dollar bill and you needto get gas for your car. If gas is $3.35 agallon and your car gets 24 miles per

gallon, how many miles will you be able todrive your car on twenty dollars? Try it!

(Mathematically all you do is: 20 x 1 3.35 x 24 1 = 143.28 )



Example Problem #4

• Try this expanded version of the previousproblem.

• You have a twenty dollar bill and you needto get gas for your car. Gas currently costs$3.35 a gallon and your car averages 24miles a gallon. If you drive, on average,

7.1 miles a day, how many weeks will yoube able to drive on a twenty dollar fill-up?




• $ 20.00 1 gallon 24 miles 1 day 1 week

$ 3.35 1 gallon 7.1 miles 7 days

= 2.88 weeks

(Mathematically : 20 x 1 3.35 x 24 1 x 1 7.1 x 1 7 = 2.88 )




• So you can have a simple 1 step problemor a more complex multiple step problem.Either way, the set-up of the problemnever changes.

• You can even do problems where youdon’t even understand what the units are

or what they mean. Try the next problem.



Example Problem #5

• Peter Piper picked a peck of pickled peppers... Or so the rhymegoes. (What in the world is a peck?)

• If Peter Piper picked 83 pecks of pickled

peppers, how many barrels is this?

• You need help for this one. As long as you

have information (conversion factors) youcan solve this ridiculous problem.




• Use this info: A peck is 8 dry quarts: a bushel is 4 pecks or 32 dry quarts; a barrel is 105 dry quarts.

• WHAT?! Rewrite them as conversion factors if the infois not given to you that way.

8 dry quarts or 1 peck .1 peck 8 dry quarts

4 pecks or 1 bushel1 bushel 4 pecks

32 dry quarts or 1 bushel .1 bushel 32 dry quarts

105 dry quarts or 1 barrel .1 barrel 105 dry quarts



• 83 pecks 8 dry quarts 1 barrel1 peck 105 dry quarts

Example Problem #5

• Pick the conversion factors that will helpget to the answer. 83 pecks 1 barrel.Hint: Look for units that will cancel each other.

= 6.3 barrels

(Mathematically : 83 x 8 1 x 1 105 = 6.3 )



Review

• Dimensional Analysis (DA) is a methodused to convert from one unit system toanother. In other words a math problem.

• Dimensional Analysis uses Conversionfactors . Two terms that describe the same or equivalent

“amounts” of what we are interested in.

• All DA problems are set the same way.Which makes it nice because you can doproblems where you don’t even understandwhat the units are or what they mean.

dimensional analysis powerpoint

Documents