Review SI Units
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Time:
Mass:
Distance:
Temperature:
Charge:
Important Secondary Quantities
• Force:
• Energy (work):
• Power:
Questions:
• What is a kilowatt-hour?
• Where is it used?
In-Class Activity #1
A study of households and businesses in
the Boston, Massachusetts area found
that air conditioning units were used for
an average of 4600 hours per year.
Determine the total annual cost of
electricity required to operate a 10,000
Btu/hr air conditioning unit in the Boston
area if the electric rate is $ 0.071/kWhr.
Notation
• The units of variables will be referred to by
putting the variables in brackets
• Consider the equation: v-vo = a·t
• What is the equation saying?
• [v] will refer to the dimension of v
[t] will refer to the dimension of t
• In order to determine the units of an unknown
(say a), we need to be able to write equations
in terms of units, e.g.
v  vo   at 
Rules of Homogeneity
• Definition: An equation is said to be
dimensionally homogeneous if all terms
separated by plus and minus and equal sign
have the same dimension.
• Consider the previous example:
v  vo   at 
• In order to be homogeneous,
v  vo   a  t 
Rule 1
• If the dimensional quantities are replaced
by their primary units the equation should
reduce to an identity.
• In our example, what are the primary units?
v  vo   a  t 
Rule 2
• Dimensions do NOT add or subtract. In
order to add or subtract variables, they
must have the SAME units.
• In our example, [v] = [vo]. If you subtract
them you have no units on the left
Rule 3
• Dimensions DO multiply and divide.
In-Class Activities #2, #3
• If P1 = 400 W and P2 = 12 Btu/minute, what is
P1+P2 in SI units?
• Which, if any, of the following are correct for
acceleration units:
m
m
a) m/s/s b) s c)
ss
s
m 1

d)
s s
Exercises
• A relationship between Force F in (N),
distance x in (m), mass M in (kg) and
speed v in (m/s) is suggested as
1
2
Fx  Mv
2
Is this equation dimensionally
homogeneous?
Exercises (cont)
• A relationship between Force F in (N),
time t in (s), mass M in (kg) and speed v in
(m/s) is suggested as:
Ft2 = Mv
Is this equation dimensionally
homogeneous?
Prefixes
• As you’ve already seen, we’ll deal with
both very large numbers and very small
numbers.
• We will use scientific notation and/or
engineering prefixes to represent these
numbers
Refresher: common prefixes
T
G
M
k
tera
giga
mega
kilo
1012
109
106
103
d
c
m
μ
n
p
deci
centi
milli
micro
nano
pico
10-1
10-2
10-3
10-6
10-9
10-12
Examples in Engineering
Notation
• 2x103 Volts =
• .00045 A =
• 1.3x10-6 C =
• 10x107 Hz =
Some Quantities Must Be
Dimensionless
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In Calculus you will see eax
ax must be dimensionless (no units) 1/1
Example, if x is in m, a is in 1/m
If x is in s, a is in 1/s
• Some dimensionless quantities need units
to make sense
• Example if v = vo e-at
v and vo have units of volts
Class Activity
In an electrical circuit the current i(t) in A
(Amperes) changes with time t in s
according to the function i(t) = e-2t.
a) How could this function be dimensionally
consistent knowing that the exponential
function is always dimensionless (RHS),
and that i(t) is in A on the LHS?
b) What are the units of the constant 2 in the
exponential function?
More on Units
• Remember x = cos(θ)?
• What are the units of θ?
Angular frequency
• What are the units of ω in y(t) = cos(ωt) if t
has units s?
• ω is called the angular frequency
• ω = 2pf