DIMENSIONS AND UNITS
Dimensions are basic concepts of physical
measurements such as:
– Length = [L]
– Time = [T]
– Mass = [M]
– Temperature = [θ]
Units are terms that precede and describe the
Classification of dimensions
Fundamental or basic dimension
– dimensions that are measured independently
and enough to express essential physical
– dimensions that are products or quotients of
Systems of units
SI (Le Systeme Internationale d’Unites)
• Simple system because fewer names are
associated with the dimensions.
• The current metric system.
• Use prefixes (e.g., c, M, n, m) and are factors of
AE (American Engineering) system
• Deeply rooted in the United States.
• Other names of this system are English, U.S.
Customary or Imperial System.
Countries that do not use SI:
Liberia, Myanmar and United States
SI dimensions and units
Source: Himmelblau, D.M. & Riggs, J.B., 2004
AE dimensions and units
Source: Himmelblau, D.M. & Riggs, J.B., 2004
Some important tips about units:
– Uppercase and lowercase letters should be strictly
followed, e.g. K (kelvin), Pa (pascal), L (liter).
– Unit abbreviations have the same form for both singular
and plural and NOT followed by a period (.) except for
– Multiplication of two or more units will combine those two
or more units together separated by a period (.) e.g. m.s.
– Hyphen (-) should NOT be used in combination of units.
– Dot (.) in multiplication of numbers should be AVOIDED
such as 2 . 5.
– Commas in numbers (e.g. 100,000) should also be
Mathematical operations with units
• Addition, subtraction, equality
• Add, subtract, or equate numerical quantities only if
they are of the same units.
• E.g., 5 kg + 10 J are not of the same units, thus
cannot be added.
• E.g., 10 lb - 10 g can be subtracted only after the
units have been converted to be same units.
• Multiplication and division
• Multiplication and division can be done on unlike units
but cannot be cancelled or merged if they are
• E.g., 200 (kg)(m)/(s2) cannot be cancelled or
merged because the units are different from each
Handling mathematical operations: sin, cos, log and e
– The variable that the mathematical operation is applied
on must be converted to dimensionless form first.
D = 24.5 – 24.3e-0.31t t < 150 s
» where D is in meters (m) and t is in time (s). What is the
units of the constants 24.5 and 0.31 respectively?
» The unit of 24.5 is meter (m) and the unit of 0.31 must be
Conversion of units and conversion factors
• As a future scientist, technologist, or engineer, you must
pay close attention to your units.
• The procedure for converting a set of units to another is
by multiplying the number and its units to the ratio
required (a.k.a. conversion factor)
• Grid method is a simple method to use to avoid
confusion when converting units.
• Examples of conversion factors:
» 1 m = 100 cm 1 m / 100 cm or 100 cm / 1 m
» 4.45 N = 1lbf 4.45 N / 1 lbf or 1 lbf / 4.45 N
Convert from 328 ft/s to mi/h.
You need to know the required conversion factors such as,
• 1 mi = 5280 ft
• 1 min = 60 s
• 1 h = 60 min
Using the grid method,
1 h1 min5280 fts
60 min60 s1 mi328 ft
= 234 mi/h
Convert from 452 cm/s2 to m/min2.
You need to know the required conversion factors such as:
• 1 m = 100 cm
• 1 min = 60 s
(1 min)2100 cms2
(60 s)21 m452 cm
= 16272 m/min2
Pound mass (lbm) and pound force (lbf)
Newton’s 2nd law (SI system) for weight
F = Cma
Where, F = force
C = constant
m = mass
a = acceleration
• In the SI system, force of 1 N is where 1 kg is
accelerated at 9.8 m/s2; C has to be 9.8 (N)/[(kg)(m)/s2]
9.8 m1 kg1N
F = = 9.8 N
Newton’s 2nd law (AE system) for weight
•lbf and lbm can be the same value if it is at Earth’s surface
•Mass of 1 lbm is accelerated at g ft/s2 (= 32.2 ft/s2)
• is a constant
•lbf and lbm are not the same units
•1 lbf ≈ 4.44822 N
g ft1 lbm1(lbf)(s2)
F = = 1 lbf
lb f s
•A basic principle states that equations must be dimensionally consistent.
•Using van der Waal’s equation as an example,
What are the dimensions of a and b?
– ‘a’ has the units (pressure)(volume)2
– ‘b’ has the same units as ‘V’ [volume]
•There are some variables or group of variables that do not have a net unit.
These are called non-dimensional or dimensionless variables, for example,
Any meaningful value have 3 types of information
associated with it:
1. the magnitude of the variable being measured.
2. its units.
3. an estimate of its uncertainty.
• The number 140.06 have 5 significant figures
• 140.06 lies in the uncertainty interval of
• 140.06 ± 0.005
• From 140.055 to 140.065
• If a number is displayed as 130.000, it means that the
number is more accurate since it contains 6 significant
Multiplying or dividing numbers
• A very important tip is to keep the final answer the lowest
number of significant figures when multiplying or
40.392 × 87.0345 ÷ 0.32 = 11000 (2 s.f.)
Adding and subtracting numbers
• When adding or subtracting, the significant figure that
should be kept in the final answer must be determined
by the largest error interval. For example,
125.8 + 0.045 = ?
Error intervals of 125.8 and 0.045 are:
• 125.8 ± 0.05 and 0.045 ± 0.0005
• The larger error of 125.8 obscures the error of 0.045
• Thus,125.8 + 0.045 = 125.845 = 125.8 (4 s.f.)
• This is because the final summation should account for
only the larger error of 0.1 from 125.8
Something to think about,
• Avoid increasing the precision (number of significant
figures) of the final answer when compared to the
values used in the calculations.
• One or two figures can be used in the intermediate
• Numbers such as 1 kg or 20 cm can be assumed that
its number of significant figures are high (such as
1.000 kg or 20.000 cm). They are called PURE or
DEFINED numbers, such as 3 cars or 2 apples, and
Calculate the following, giving the accurate number of s.f. in
each final answer.
Tip: Keep the same number of decimal places as the number with the least
amount of decimal places.
• 1.421 + 0.4372 =
• 0.0241 + 0.11 =
• 0.14 + 1.2243 =
• 760.0 + 0.011 =
• 1.0123 – 0.002 =
• 123.69 – 20.1 =
• 463.231 – 14.0 =
• 47.2 – 0.01 =