Dimensional Analysis 12.ppt

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Dimensional Analysis

Method

Follow along on your Dimensional Analysis Worksheet and do the example problem with the

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2

Which is bigger?

₁₂₀ ₅

₂₁

Pennies Quarters Nickels

Answer = 5

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A physical quantity must include:

Number

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4

Calculation Corner: Unit Conversion

1 foot = 12 inches

1 foot

12 inches

= 1

12 inches

1 foot

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5

The density of aluminum is 2.70 g/mL. What does this mean?

How many significant figures are in 2.70 g and 1 mL?

A rule of thumb for conversion factors that have decimal values: They are not unlimited and you need to look at the number of sig figs

Examples:

2.70 g = 1 mL

2.70 g = 1.00 mL 3 sig figs 3 sig figs

1 m =39.37 inches 2.54 cm = 1 inch

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What is Dimensional Analysis?

_{Dimension – means Unit}

_{Analysis – means Problem Solving}

Math using units!

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Setup

1. Write down given 2. Set up workspace

3. Write down units of answer

How many dimes are in 30 cents?

30 cents

₌

_dimes

Start writing now!

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Solution

4. Do I want to keep it? If you don’t, write the unit on the bottom of the next line.

5. What can I change it into? Write that unit on top and then include the numbers.

6. Repeat until the answer to #4 is yes!

30 cents

₌

_dimes

cents dime 1

10

Step 4 Step 5

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Calculate

7. Everything on top is multiplied and everything on the bottom is divided.

8. Plug the numbers in the calculator and let it do all the work!

30 cents

₌

_dimes

cents dime 1

10

3 dimes



( 10) ( 30) ( 1)

Step 7

Step 8

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More info – Step 4

_{Step 4 –}_{Do I want to keep it?}

The it refers only to the unit. Don’t recopy the number!

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More info - Step 5

_{Step 5 -What can I change it into?}

This really means “what do I know it equals”. In the example, we wrote dime on top because we know that 10 cents = 1 dime. We also know

100 cents = 1 dollar, but while we could write dollar on top, it wouldn’t help us solve the

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More info – Step 5 continued

In this step, you are finishing what you started in step 4. You are actually completing what is termed a “conversion factor” (a ratio of two

equivalent values). All conversion factors then equal 1.

60 seconds = 1 minute can be written:

onds ute or ute onds sec 60 min 1 min 1 sec 60

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Setup

1. Write down given. 2. Set up workspace.

3. Write down units of answer.

How many cm are in 4.75 feet?

4.75 feet

=

cm

Step 2 _{Step 3}

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Solution

4.75 feet

=

cm

foot inches 12

1

Step 4 Step 5

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Solution

4.75 feet

=

cm

foot inches 12

1

Step 6 –inches cancel and we are left with cm so “yes” and we can now calculate answer

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Calculate

7. Everything on top is multiplied and everything on the bottom is divided.

8.Plug the numbers in the calculator and let it do all the

work!

9.Round to correct sig figs.

cm

145 cm



( 1) ( 1)

( 2.54) ( 4.75) ( 12)

Step 7 Step 8 4.75 feet

=

foot inches 12 1 2.54 cm 1 inch

12 inches = 1 ft is a

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Homework

Use this technique to complete the

Dimensional Analysis worksheet!

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1. Multi-step problems

=

2.5 wks _s

wks days 1 7 days hours 1 24 hours min 1 60 _1512000 min s 1 60 1500000 s

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2. Two step metric conversions

(prefix unit to a prefix unit)

=

17.8 cm _km

cm m 100 1

m km 1000

1 _.000178

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Measured dimensions of a rectangle:

Find area of rectangle. A = L . W

= (9.70 cm)(4.25 cm) length (L) = 9.70 cm

width (W) = 4.25 cm

L

W =

Square and Cubic U nit Conversions Example Problem

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Convert 41.2 cm2 to m2.

100 cm 1 m

(

______

)

X m2 = 41.2 cm2

X m2 = 41.2 cm.cm

Recall that… 41.2 cm2 = 41.2 cm.cm

100 cm 1 m

(

______

)

X m2 = 41.2 cm2 = 0.412 m2

= 0.412 cm.m

WRONG!

(

______

)

100 cm 1 m

= 0.00412 m2

(

______

)

100 cm

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Measured dimensions of a rectangular solid:

Find volume of solid.

L W

H Length = 15.2 cm

Width = 3.7 cm Height = 8.6 cm

V = L . W . H

= (15.2 cm)(3.7 cm)(8.6 cm)

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3. Square and cubic unit conversions

How many m3 are in 3 cm3?

X m3 = 3 cm3 = .000003_m₃

100 cm

1 m 3

(

_____

)

X m3 = 3 cm3

= X m3 = 3

100 cm 1 m

(

_____

)

100 cm 1 m

(

_____

)

(

_____

)

₌ or

cm.cm.cm

1 m

1000000 cm

(

_________

)

3 3

3 x 10-6 m3

or

32 cm

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4. Complex Units

(the hidden equality)

What does 60 mi/hr (this is a complex unit) really mean?

60 miles = 1 hr

hr miles

1 60

miles hr

60

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4. Complex Units

a

. Using as a conversion factor

The density of gold is 19.3 g/mL. What is the mass if the volume is 244.8 L?

=

244.8 L _g

L mL 1

1000

mL g 1

19.3 _4.724640_₁₀6

19.3 g = 1 mL

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4. Used as a Complex Units

b. Using as a given

The density of gold is 19.3 g/mL, what is it in cg/kL?

=

19.3 g = 1 mL

cg kL 19.3 g

1 mL _g

cg 100 1 mL L 1000 1 L kL 1000 1