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Dimensions, Units, and

Conversions

Introduction to Engineering

Objectives

• Explain the difference between dimensions and

units.

• Check for dimensional homogeneity.
• Explain SI unit prefixes.
• Convert between SI and U.S. Customary units.
• Explain the difference between mass and weight.

Assignment: Handout or visit website.

Difference Between Dimensions

and Units

• Dimensions are divided into fundamental

and derived. Fundamental are the most basic or elementary dimensions necessary to describe the physical state of an object. Derived dimensions are defined based upon scientific and engineering equations, and are a combination of fundamental dimensions.

Fundamental and Derived Dimensions

Dimensional Analysis

• Equations in Science and Engineering must be dimensionally homogeneous, in other words, the dimensions on each side of the equation should be the same when dimensions ( not units ) are substituted for the variables and constants.
• For example, if you are calculating velocity from the distance traveled in an elapsed time, the dimensions on either side should be equal, i.e. ,

Velocity = Distance traveled / Elapsed time Distance traveled = Length (L) Elapsed time = Time (T) Velocity = Length / Time = L / T

Unit Systems

• Systems of units differ in the treatment of

mass and force.

• In the SI system, mass was chosen as the

third fundamental dimension and force is

a derived unit.

• In the English system, force was chosen as

the third fundamental dimension and

mass is a derived unit.

Base Units

Derived Units

The International System of Units

• To avoid very small or

very large numbers in

the SI system of units,

unit prefixes have

been developed based

on power of ten.

Unit Systems

• Fundamental and some important derived

dimensions for the three common systems of units.

Unit Systems and Conversions Exact Conversions

The internet provides valuable resources that can be used to obtain a variety of different conversion factors or completely carry out the conversions for you. Please refer to the following website:

Unit Systems and Conversions Example

• The employment of the information given in the preceding tables allows for ease of conversion between different units.
• For example, if you are traveling at a speed of 65

miles per hour (mi/hr or mph) and wish to know your speed in feet per second (ft/s) and in meters per second (m/s) you would have to carry out the following conversions:

65

mi hr

5280

ft mi

⋅

1 60

hr min

⋅

1 60

min s

⋅ 95.

ft s

:= ⋅

ft s

1

m ft

⋅ 29.

m s

:= ⋅

Quiz

• Carry out the following conversions:

a) 125 days to seconds

b) 16 lbm/ft^3 to kg/m^3

c) 75 slug/min to kg/s

d) 15 ft^3 to gallons

Quiz Solutions

( 125 day ⋅ ) 24 hr day

  ⋅ 

  

⋅ 60 min hr

  ⋅ 

  

⋅ 60 sec min

  ⋅ 

  

⋅ :=1.08 10 ⋅ 7 ⋅sec

16

lbm

ft 3

⋅

  

  

1

kg lbm

  ⋅ 

  

⋅ 3. ft m

  ⋅ 

  

3 ⋅ 256. kg

m^3

:= ⋅

a)

b)

Notice that the (ft/m) part is cubed because we cannot cancel out ft 3 with just ft, remember , the dimensions must be the same.