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DIMENSIONS AND UNITS

Definition: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 dimensions.

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Classification of dimensions

Definition:Fundamental or basic dimension

– dimensions that are measured independently and enough to express essential physical quantities

Derived dimensions– dimensions that are products or quotients of

fundamental dimensions

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Systems of units

SI (Le Systeme Internationale d’Unites) system

• 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

10.

AE (American Engineering) system• Deeply rooted in the United States. • Other names of this system are English, U.S.

Customary or Imperial System.

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Countries that do not use SI: Liberia, Myanmar and United States

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SI dimensions and units

Source: Himmelblau, D.M. & Riggs, J.B., 2004

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AE dimensions and units

Source: Himmelblau, D.M. & Riggs, J.B., 2004

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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 inches (in.).

– 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

AVOIDED.

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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 different.• E.g., 200 (kg)(m)/(s2) cannot be cancelled or

merged because the units are different from each other

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Handling mathematical operations: sin, cos, log and e

– The variable that the mathematical operation is applied on must be converted to dimensionless form first.

Example 1

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 s-1.

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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

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Example 2 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

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Example 3

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

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Pound mass (lbm) and pound force (lbf)

Newton’s 2nd law (SI system) for weight

F = CmaWhere, F = force

C = constantm = massa = 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]

s2

s2(kg)(m)

9.8 m1 kg1N

F = = 9.8 N

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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

s232.174(lbm)(ft)

g ft1 lbm1(lbf)(s2)F = = 1 lbf

32.174lbm ft

lb f s2

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Dimensional consistency

•A basic principle states that equations must be dimensionally consistent.•Using van der Waal’s equation as an example,

Example 4What are the dimensions of a and b?

– ‘a’ has the units (pressure)(volume)2

– ‘b’ has the same units as ‘V’ [volume]

Dimensionless numbers•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,

gcm3s

(cm)(s)gcmcm

Dv

N RE

RTbVV

ap

2

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Significant figures

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.

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Example 5

• 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 figures.

Multiplying or dividing numbers• A very important tip is to keep the final answer the lowest

number of significant figures when multiplying or dividing.

Example 6

40.392 × 87.0345 ÷ 0.32 = 11000 (2 s.f.)

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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,

Example 7125.8 + 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

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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 calculations.

• 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 sometimes dimensionless.

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Example 8

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 =