Dimensions, Units, and
Conversions
Introduction to Mechanical Engineering
The University of Texas-Pan American
College of Science and 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
 Why are dimensions and units important?
 Dimensions are used to describe objects
and actions. The three most basic
dimensions are length, time, and mass.
 Units are used to establish the size or
magnitude of a dimension. Must be based
on some convention with standards
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
Difference Between Dimensions
and Units
 Dimensions are fundamental, unchanging
characteristics or properties of an object.
 Units on the other hand are arbitrary; they
can be changed by the vote of a governing
body.
 History of Units – cubit, meridian mile,
foot, etc…
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
Dimensional Analysis – Example 1
The Reynolds number is given by
Re 
Dv

where D = pipe diameter, v = fluid velocity, ρ = fluid
density, and µ = viscosity (M/LT). Show that the Reynolds
number is dimensionless.
 L  M 
L  3 
 T  L   L L M LT  Dimensionl ess
3
M
T
M
L




 LT 
Dimensional Analysis – Example 2
The pressure in a column of fluid is given by
P   gh
where P = pressure, ρ = fluid density, g =acceleration of
gravity, and h = height of fluid column. Is this equation
dimensionally homogeneous?
M
M L
M

L

2
3
2
2
LT
L T
LT
YES
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.
The International System of Units
 SI units are derived into three classes: base
units (seven), derived units, and
supplementary units (two).
Base Units
Derived Units
Supplementary Units
 Radian is equal to the angle between two radii of
a circle that cut off a piece of the circumference
whose length is equal to the length of the radius.
 Steradian is equal to the solid angle which cuts
off, on the surface of a sphere, an area equal to
the area of a square whose sides are the same
length as the radius of the sphere.
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
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:
http://www.onlineconversion.com/
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    5280 ft    1 hr    1  min   95.333 ft





hr
mi
60
min
60
s
s





 95.333 ft    1  m   29.056 m

  3.281 ft 
s
s



Mass & Weight
 The mass of an object is constant.
 Weight is the force required to lift or support an
object in a gravitational field or an acceleration
field.
 Acceleration of gravity changes with location.
 For example, on the Moon, your mass would be
the same as here on Earth, yet your weight
would be less due to the lower gravitational
acceleration present on the Moon.
Open Forum
Quiz
 Carry out the following conversions:
a) 125 days to seconds
b) 16 lbm/ft3 to kg/m3
c) 75 slug/min to kg/s
d) 15 ft3 to gallons
Quiz Solutions
a)
( 125 day )  
 24

   60 min    60 sec   1.08 107 sec



day  
hr  
min 
hr
b)
3
lb m 

1
kg
ft
   3.281   256.336 kg

16


 2.2046 lb  
3 
m
3



m


ft
m


Notice that the (ft/m) part is cubed because we cannot
cancel out ft3 with just ft, remember, the dimensions
must be the same.
Quiz Solutions
c)
lb
 75 slug    32.174 m    1  kg    1  min   18.243 kg
 min 


slug   2.2046 lb m   60 sec 
sec




d)
15 ft3  7.48052 gallons   112.208 gallons

ft
3
