UNIT DAN DIMENSI

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DIMENSI

Suatu gambaran jenis dari kuantitas fisik

Contoh:

- Panjang - gaya- luas - massa- volume - kecepatan- waktu - suhu

- dan lain-lain

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UNIT

Sesuatu yang digunakan untuk menyatakan ukuran atau kuantitas dari suatu dimensi

Ada bermacam-macam SISTEM UNIT yang digunakan :

- British Unit

- American Unit

- Engineering Unit

-Yang umum digunakan adalah Unit SISTEM

INTERNASIONAL (SI) 3

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Some Tools for Measurement

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Standard Kilogram at Sèvres

SISTEM PENGUKURAN

SYSTEM USEDIMENSI

LENGTH MASS TIME TEMP FORCE ENERGY

ENGLISH ABSOLUTE

scientific foot (ft) Pound mass (lbm)

sec. °F Poundal BTU ft (pou-ndal)

BRITISH ENG.

INDUSTRI foot (ft) slug sec. °F pound force (lbf)

BTU ft (lbf)

AMERICAN ENG.

US. INDUSTRY

foot (ft) pound mass (lbm)

sec. °F lbf BTUft (lbf)

METRIC

CGS Scientific cm gram sec °C dyne Calori, erg

MKS industri m kg sec °C kgf Kilocalori, joule

SIuniversal m kg sec °C newton joule

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Unit Dasar

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QUANTITY UNIT NAME

UNIT DIMENSIONS

Basic UnitMass Kilogram kg MLength, Diameter Meter m LTime Second s TTemperature Kelvin K θ

Unit turunan

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QUANTITY UNIT NAME

UNIT DIMENSIONS

Derived unit with a special nameForce Newton (N) kg m s-2 MLT-2

Pressure Pascal (Pa) Nm-2 ML-1T-2

Energy Joule (J) Nm ML2T-2

Power Watt (W) Js-1 ML2T-3

Frequency Hertz (Hz) Hz T-1

Absorbed dose of ionising radiation

Gray (Gy) Jkg-1 L2T-2

Unit turunan

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QUANTITY UNIT NAME

UNIT DIMENSIONS

Derived unit without a special nameArea m2 L2

Volume m3 L3

Density kgm-3 ML-3

Dynamic viscosity N s m-2 ML-1T-1

Kinematic viscosity M2s-1 L2T-1

Enthalpy Jkg-1 L2T-2

Specific heat Jkg-1K-1

(or oC)LT-2θ-1

TEHNIK KONVERSI

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Ada beberapa cara untuk mengkonversi dari unit yang

satu ke unit yang lain, dalam satu sistem atau antar

sistem :

1.Dengan Tabel

2. Dengan simbol Prefix

3. Dengan menghitung

Tabel Konversi

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QUANTITY BRITISH SYSTEM SILength 1 ft 0,3048 mTime 1 h 3,6 ksArea 1 ft2 0,09290 m2

Volume 1 ft3 0,02832 m3

Mass 1 lb 0,4536 kgDensity 1 lb ft-3 16,019 kg m-3

Force 1 lbf 4,4482 NEnergy 1 Btu 1055,1 J

1 cal 4,1868 JPressure 1 lbf in-2 6894,8 Pa

1 atm 1,0133 x 105 Pa1 torr 133,32 Pa

Power 1 Btu h-1 0,29307 W1 hp 745,70 W

Velocity 1 ft s-1 0,3048 m s-1

Dynamic viscosity 1 P (poise) 0,1 N s m-2

Kinematic viscosity 1 St (stokes) 10-4 m2 s-1

Specific heat 1 Btu lb-1 oF-1 4,1868 kJ kg-1 K-1

Thermal conductivity 1 Btu h-1 ft-1 oF-1 1,7303 W m-1 K-1

Heat transfer coefficient 1 Btu h-1 ft-2 oF-1 5,6783 W m-2 K-1

Mass transfer coefficient 1 lb ft-2 s-1 1,3563 g m-2 s-1

Temperature oF oC9

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Sistem konversi dengan PREFIX

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PREFIX MULTIPLE SIMBOLTERA 1012 TGIGA 109 GMEGA 106 MKILO 1000 (103) kMILLI 10-3 m

MICRO 10-6 μNANO 10-9 nPICO 10-12 p

FEMTO 10-15 f

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SI System of UnitsForce = (mass) (acceleration)

SI System of Units: Force

Force = ma

= Newton

= N

2s

mkg

SI System of Units: Stress/Pressure

Pressure = Force / Area

= Pascal = Pa

2

2

2 m

s/mkg

m

N

2sm

kg

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U.S. Customary System of Units (USCS)

Fundamental Dimension Base Unit

length [L]

force [F]

time [T]

foot (ft)

pound (lb)

second (sec)

Derived Dimension Unit Definition

mass [FT2/L] slug lbf sec2/ft

USCS: Force = (mass)*(acceleration)

2f ft/secslug1lb1

F = ma

W = mg

American Engineering System

Note, there is a problem when we use the same unit (“pound”, meaning lbf and lbm) to describe two different dimensions.

Newton's Second Law: F = ma

1 lbf = 1 lbm ft/s2 ??? NO!!! Must have consistency of units.

Consistency of Units

Principle of consistency of units: units on the left side of an equation must be the

same as those on the right side of an equation dimensional homogeneity

AES and Newton’s Law

Must maintain dimensional homogeneity:

Now we have lbf = lbf

cg

maF

2f

mc sec lb

ftlb32.174

g

KONSTANTA DIMENSIONAL (gc)

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Muncul karena adanya gaya gravitasi bumi

1 lb dalam berat diatas permukaan bumi disebabkan oleh :

1 lb massa yang dipengaruhi oleh gaya gravitasi

2sec1

ftxlblb mf

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xft

xlblb mf 2sec Dimensional konstan

Cmf g

xft

xlblb1

sec2

2secf

mc lb

ftlbg

Untuk menghasilkan konsistensi dalam unit

dibutuhkan konstanta

Gravitasi bumi = 2sec174,32

ft

2sec174,32

f

mc lb

ftlbg

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Force and Weight

Be sure you understand the difference between lbf and lbm

Be sure you understand the difference between the physical constant g, and the conversion factor gc.

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FORCE, WEIGHT AND MASS Force is proportional to product of mass and

acceleration and is defined using derived units to equal the natural units;

1 Newton (N) = 1 kg.m/s2

1 dyne = 1 g.cm/s2

1 Ibf = 32.174 Ibm.ft/s2 Weight of an object is force exerted on the object by

gravitational attraction of the earth i.e. force of gravity, g.

To convert a force from a derived force unit to a natural unit, a conversion factor, gc must be used.

A ratio of gravitational acceleration, g to gc may be used for most conversions between mass and weight.

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FORCE, WEIGHT AND MASS

1. F = ma /gc : W = mg /gc

kg.m/s2 g.cm/s2 Ibm.ft/s2 2. gc = 1 --------- = 1 --------- = 32.174 -----------

N dyne Ibf 3. g = 9.8066 m/s2 ===> g/gc = 9.8066 N/kg g = 980.66 cm/s2 ===> g/gc = 980.66 dyne/g g = 32.174 ft/s2 ===> g/gc = 1 Ibf/Ibm

4. Example: Water has a density of 62.4 Ibm/ft3. How much does 2.000 ft3 of water weigh?

Dimensional Analysis

Dimensions & units can be treated algebraically.

Variable from Eq.

x m t v=(xf-xi)/t

a=(vf-vi)/t

dimension L M T L/T L/T2

Dimensional Analysis

Checking equations with dimensional analysis:

L(L/T)T=L

(L/T2)T2=L

• Each term must have same dimension• Two variables can not be added if dimensions

are different• Multiplying variables is always fine• Numbers (e.g. 1/2 or p) are dimensionless

x f xi vit 1

2at 2

Example 1.1

Check the equation for dimensional consistency:

2

2

2

)/(1mc

cv

mcmgh

Here, m is a mass, g is an acceleration,c is a velocity, h is a length

Example 1.2

L3/(MT2)

Consider the equation:

Where m and M are masses, r is a radius andv is a velocity.What are the dimensions of G ?

mv2

rG

Mm

r2

Example 1.3Given “x” has dimensions of distance, “u” has dimensions of velocity, “m” has dimensions of mass and “g” has dimensions of acceleration.

Is this equation dimensionally valid?

Yes

Is this equation dimensionally valid?

No

x (4 / 3)ut

1 (2gt 2 / x)

x vt

1 mgt 2

Units vs. Dimensions

Dimensions: L, T, M, L/T … Units: m, mm, cm, kg, g, mg, s, hr, years … When equation is all algebra: check

dimensions When numbers are inserted: check units Units obey same rules as dimensions:

Never add terms with different units Angles are dimensionless but have units

(degrees or radians) In physics sin(Y) or cos(Y) never occur

unless Y is dimensionless

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Example The density of a fluid is given by the empirical

equation

ρ = 1.13 exp(1.2 x 10-10 P)Where ρ = density in g/cm3

P = pressure in N/m2

a) What are the units of 1.13 and 1.2 x 10-10?b) Derive the formula for r(lbm/ft3) as a function of P (lbf/in2)

A column of mercury is 3 mm in diameter x 72 cm high. If the density of mercury is 13.6 g/cm3, what is its weight in N. What is its weight in lbf? What is its mass in lbm?

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Example

The Reynolds number is the dimensionless quantity that occurs frequently in the analysis of the flow of fluids. For flow in pipes it is defined as DVρ/μ, where D is the pipe diameter, V is the fluid velocity, ρ is the fluid density, and μ is the fluid viscosity. For a particular system having D = 4.0 cm, V = 10.0 ft/s, r = 0.700 g/cm3, and μ = 0.18 centipoise (cP) (where 1 cP = 6.72 x 10-4 lbm/ft.s). Calculate the Reynolds number.