Dimensional Analysis Made Easy

DESN 100

Christine GriffithSpring 2009

So what is dimensional analysis?

• Dimensional analysis is the process used by scientists, mathematicians, engineers to change from one unit of measure to another

An example problem

• Convert 60 mph into feet per second

mile

fthour

hour

miles

1

5280

sec60

min1

min60

160

Notice that the original units of miles per hour are multiplied by fractions that are equal to 1. 1 hour = 60 minutes5280ft=1mile1 minute = 60 seconds

Cancelling on the diagonal

mile

fthour

hour

miles

1

5280

sec60

min1

min60

160

Hour on top cancels with hour on the bottomMinute on top cancels with minute on the bottomMiles on top cancels with mile on the bottom

Finish the problem…

fpsft

ftft

88sec

88sec60

5280

1

5280

sec60

1

60

1

1

60

Now notice that the cancelled units turned into 1’sThe 60’s also cancel

Procedure summary

1. Write the original number with dimensions as a fraction. Example: 60 miles/hour

2. Multiply by fractions with equivalent units in the top and bottom of the fraction. Example: 1 hour/60 minutes.

3. Make sure the units cancel top and bottom until you are left with the units you want

Another problem to practice

• Convert 100 ft/s² to mm/s²

2

2

30480

1

10

1

54.2

1

12100

s

mm

cm

mm

in

cm

ft

in

s

ft

A last “tricky” problem

• Sometimes the units we need to change are raised to a power…

• Example: Convert 10 psi (1b/in²) into N/mm²

• N stands for Newtons, the unit of force in SI units

Recall: (a/b)²=a/b * a/b

2

2

2

2

0689.040.25

1

40.25

1

1

448.410

40.25

1

1

448.410

mm

Nmm

in

mm

in

lb

N

in

lb

mm

in

lb

N

in

lb