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CHAPTER 5

Dimensions & Units

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• State basic quantities and their respective SI units

• State derived quantities and their respective SI units.

• Realize the existence and importance of SI, international system of standard measurements.

• State and use dimensions for base quantities

• Deduce the correct dimension for a certain derived quantity and its equation of physics

OBJECTIVES

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5.1 Introduction5.2 Physical Quantity5.3 Units5.4 SI Units5.5 Rules for SI units5.6 USCS5.7 Dimensions

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Introduction• Rapid increase in global communication and the

development of technology require a universal measurement system.

• The standard currently accepted in most nations is the international metric system or International System of units (SI system).

• SI units are modification and refinement of earlier metric system (MKS).

• This chapter deals with the physical quantities, SI Units and the differences between dimensions and units.

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Physical Quantities “Measured Quantity”

•For instance; length, time, temperature, force, volume, velocity and so on.

•Physical Quantity can be divided into 2 types

Fundamental or Basic (quantity that cannot be derived from any other

quantity)

Derived(quantity that is derived from the combination of several basic quantities by mathematical operation.)

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Fundamental Physical Quantity

Physical Quantity Unit SI SymbolMass Kilogram, kg m

Length meter, m l

Time second, s t

Temperature Kelvin, K T

Electric current ampere, A I

Amount of substance mol, mole mol

Luminous intensity candela, cd I

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Derived Physical Quantity

Physical Quantity Derivation unit

Area m2

Volume m3

Velocity ms-1

Acceleration ms-2

Force N or kgms-2

Momentum kgms-1

Work J or kgm2s-2

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Example : Derive Quantity

time

ntdisplaceme velocitya) 1(meter)

meter per second m s(second)

time

velocity on accelerati b)

12(meter per second) m s

m s(second) s

volume

mass density c) -3

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(kilogram)kilogram per cubic meter kg m

(meter)

eleration)(mass)(acc force d) 2 -2(kg)(ms ) kg m s newton, (N)

area

force pressure e)

-2-1 -2 -2

2

(kg m s )kg m s N m pascal, (Pa)

(m)

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Units

• All physical quantities have units

• SI unit

• Prefixes are used to represent larger or smaller values.

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SI Units•Developed and maintained by the CGPM (General Conference on Weights and Measures). •Name was set in 1960.• Intended as a basis for world wide

standardization.•Currently, the system is complete and used

universally.•New SI system is divided into 3 classes of

quantity units Base Quantity Units Supplementary Quantity Units Derived Quantity Units

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Base Quantity Units

•There are seven Base units in the SI

- Length- Time- Mass- Electric current- Temperature- Amount Of substance- Luminous intensity

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SI Unit Definitions• Length 1 meter = the distance traveled by light in a vacuum

during 1/299 792 458 s (1983).• Time 1 second = duration of 9192 631 770 periods of

radiation corresponding to the transition between the two hyperfine levels of the grounds state of cesium-133 atom (1967).

The nation’s primary time standard is a cesium fountain atomic clock developed at the National Institute of Standards and Technology laboratories in Boulder, Colorado. The clock will neither gain nor lose a second in 20 million years. 

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• Temperature1 Kelvin= 1/273.16 of thermodynamic temperature of the triple point of water (1967)

• Mass 1 kg = mass of a cylinder of platinum-iridium

alloy kept by the international Bureau of Weights and Measures in France (First 1889-Third 1901)

The National Standard Kilogram No. 20, an accurate copy of the International Standard Kilogram kept at Sèvres, France, is housed under a double bell jar in a vault at the National Institute of Standards and Technology.

SI Unit Definitions

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Supplementary Units

• No longer been used since Oct. 1995, both units moved into the class of derived units

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Derived Units•Formed by combining base,

supplementary or other derived units. •Derived units with special names and

symbolsQuantity SI Unit

SymbolName Base Unit

Frequency Hz hertz s-1

Force N newton kgms-2

Pressure stress Pa pascal kgm-1s-2

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•Derived units with no special units names but are combinations of base units and units with special names.

Quantity Unit

Acceleration ms-2

Angular acceleration rads-2

Area m2

Derived Units

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Rules of Units

•Engineers should understand thoroughly and apply properly the new system.

•To apply, we have to follow the rule. - Units Symbol and Names- Multiplication and division- Numbers- Calculating with SI units

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Rules of Units & Symbols• Units symbol written in lowercase letters ; m, kg, s,

mol.. (exception: the (A, K, Hz, Pa, C) the prefixes, mega (M)

and giga (G)

• Symbols named after an individual are capitalized Eg: newton (N)

• Symbols should be used to represent units.

• An s is never added to the symbol to denote plural.

• Use space between numerical and units symbol. (exception: for degree, minutes and second for angle and for degree Celsius)

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• The symbols should be used in preference to the unit name. (10 m not 10 meters or ten meters not 10 m)

• Quantities defined by several units which are multiples, are separated by a dotEg: N = kg.m/s2 = kg.m.s-2

d

• Physical constants with several digits on either side should be written with a space between 3 digits rather than a comma, Eg: 73 569.213 427

• In calculations, represent numbers in terms of their base or derived units by converting all prefixes to powers of 10

Rules of Units & Symbols

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• Compound prefix should not be used Eg: kμs (kilo-micro-second) should be expressed as ms (milli- second) since 1 kμs = 1 (103)(10-6) s = 1 (10-

3) s = 1ms

• With exception of base unit kilogram, avoid use of prefix in the denominator of composite units Eg: Do not write N/mm but rather kN/m Also, m/mg should be expressed as Mm/kg

Rules of Units & Symbols

Numerical Calculations•Dimensional Homogeneity

Each term must be expressed in the same units

Eg: s = vt + ½ at2 where s is position in meters (m), t is time in seconds (s), v is velocity in m/s and a is acceleration in m/s2

Regardless of how the equation is evaluated, it maintains its dimensional homogeneity

All the terms of an equation can be replaced by a consistent set of units, that can be used as a partial check for algebraic manipulations of an equation

ExampleEvaluate each of the following and express with

SI units having an approximate prefix:

a. (50 mN)(6 GN), b. 45 MN3/900 Gg

Solution

222

3326

26

93

300300

10

1

10

110300

10300

1061050

650

MNNk

N

kN

N

kNN

N

NN

GNmN

(a)

kgkN

kgkN

kgN

kNN

kgN

kg

N

GgMN

/50

/1005.0

1

10

11005.0

/1005.0

10900

1045

900/45

3

33

3312

312

6

36

3

(b)

Solution

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Convert to Si Unit

(400 mm)(0.6 MN)2

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U.S. Customary System (USCS)

•Many segments of the U.S industrial complex regularly use other system.

•There are 2 systems of units other than SI that are commonly used in the United States

1. U.S Customary System2. Engineering System

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USCS

Unit Symbol

Mass slug slug

Length foot ft

Time second s

Force pound lb

USCS units are also called Imperial units. Imperial or USCS units are gradually being replaced by SI units.

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Conversion of Units

s180min1

s 60min)3(min3

131- sm33.3s3600

h1

h

m101200hkm1200

Example 1 : min s

Example 2 : km h-1 m s-1

Example 3 : cm3 m3

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3233 m101

cm1

m10cm10cm10

• In solving problems, all data must be converted to S.I units.

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Question 1 :

The density of lead is 11. 3 g cm-3 . What is the value in kilograms per cubic meter ?

.m kg1013.1m1

cm100

g1000

kg1

cm

g3.11 34

3

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Solution :

Question 2 :

How many nanoseconds does it take for light to travel 1.00 km in vacuum ?

ns.103.33s1033.3ms103.00

m 10 3618

3

Solution :

speed

distancetime

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Dimensions

Objectives• State and use dimensions for base

quantities• Deduce the correct dimension for a certain

derived quantity and its equation of physics

• Use dimensional analysis to check homogeneity and construct equations.

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•Relate physical quantity to the basic units.•Symbol: M,L,T,K,A

Basic Quantity Dimension

[mass] or [m] M

[length] or [l ] L

[time] or [t] T

[temperature] or [T] K

[electric current] or [I] A

[amount of substance] or [n]

N

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•Example: Dimension of mass=[mass]=M

•Dimensional equation: [X]=MaLbTc

Dimension for X= a dimension for mass, b dimension for length and c dimension

for temperature

•Dimensionless quantities: The quantities that do not have dimension. E.g: coefficient of friction

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•What is the dimension for a) force, b) work

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LM

T 2MLT

2 2 2MLT L ML T

a) [force] = [mass x acceleration]

b) [work] = [force x displacement]

Dimension for force are 1 for mass, 1 for length and –2 for temperature

Dimension for work are 1 for mass, 2 for length and –2 for temperature

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Dimensions for Derived Units

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LT T

LT

time

velocity onaccelerati b)

1LT time

ntdisplaceme velocity a)

Find for both derived unit

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Applications of Dimensions

•To determine the dimensions of a physical quantity.

•To check the homogeneity of an equation

•To construct an equation with given quantities

To determine the dimension of a physical quantity

The following procedure is recommended :

Identify all the physical quantities in the given equation.

Write the dimension

of each known physical quantity

Solve the equation to get the dimension of the unknown physical

quantity.

What are the dimensions of the constants a and b in the gas equation below ?

2

ap V b RTV

p = pressure V = volume R = gas constant T = temperature

Solution :

2.........(i)

ap

V

From (i) , [a] = [p][V]2

= (ML-1T-2)(L3)2

= ML5T-2

From (ii) , [b] = [V]

= L3

Example 1 : To determine the dimension of a physical quantity.

.......(ii)V b

To check the homogeneity of an equationThe following procedure is recommended :

1. Identify all the physical quantities in the equation given

2. Write the dimension ofeach known physical

quantity

3. Equate the dimensions ofthe quantities on both sides

of the equation

4. An equation is homogeneousif the dimensions on both

sides of the equation are equal

Show that v2 = u2 + 2as is homogeneousSolution :

L.H.S : [v]2 = L2T-2

R.H.S : [u]2 = L2T-2 , [2as] = L2T-2

Both sides of this equation have the same dimension homogeneous

Example 1 : To check the homogeneity of an equation

Solution :

L.H.S : [v]2 = L2T-2

R.H.S : [u] = LT-1 , [at] = (LT-2)(T) = LT-1

Dimension of the L.H.S R.H.S not homogeneous

Example 2 : To check the homogeneity of an equation

v2 = u + at

1

2

1

1

1

L1

and L1

quantity. essdimensionl is : R.H.S

L1

: L.H.S

rr

n

f

Both sides of this equation have the same dimension homogeneous

Solution :

Example 3 : To check the homogeneity of an equation

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11)1(

1

rrn

fShow that is dimensionally correct.

To construct an equation with given quantities

The following procedure is recommended :

Step 6 : Rewrite the equation with the correct value of indices for each quantity.

Step 5 : Determine the value of the indices.

Step 4 : Equate the indices of the dimensions on both sides of the equation.

Step 3 : Write the dimension of all the quantities on both sides of the equation.

Step 2 : Identify all the dimensionless quantities.

Step 1 : Write general equation that relates all the quantities.

Construct an expression for the period, t of a simple pendulum which shows how t is related to the mass m of the pendulum bob, the length l of the string and acceleration g due to gravity.Solution : Let t = k mx ly gz where k is a dimensionless constant ;

x, y, z are unknowns. Dimensions on both sides must be the same [t] = [k][m]x[l]y[g]z

T = MxLy(LT-2)z

T = MxLy+zT-2z

Equating the indices of M, L, T on both sides ;M : x = 0, L : y + z = 0 T : -2z = 1

z = -½ , y = ½ , x = 0

Period, t = k m0 l½g-½ org

lkt

Step 1

Step 3

Step 4

Step 2

Step 5

Step 6

Example 1 : To construct an equation with given quantities