Law Chapter 1 Part2 2011

8/7/2019 Law Chapter 1 Part2 2011

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CHAPTER 1

Dimensions


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DIMENSIONS

A dimension is a property that can bemeasured

Length, Time, Mass, Temperature

Can express units in dimensions e.g V [L3]

NOTE: SI Units are not dimensions

Equations need to be dimensionally consistent


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Quantity SI

Unit

Dimension

Mass Kilogram M

Length Meter L

Temperature K

Time s T


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3 SYSTEMS OF UNITS


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Dimensional Homogeneity

Quantities can be added/subtracted if ONLY

their units are same

Unit same, the DIMENSION of each term must be

the same.

Eg. : VELOCITY = LENGTH / TIME

(L) / (T) (L) / (T)


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So

Every valid equation must bedimensionally homogeneous:

all additive terms on both sides of the

equation must hav

e same dimensions

VALID EQUATION DIMENSIONALLY

HOMOGENEOUS

and?


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Examples:

F = ma where F = Force (N = kg.m/s2)m = mass (kg)

a = acceleration (m/s2)

F = ( M ) ( L ) / ( T )

2

, m = ( M ) , a = ( L ) / ( T )

2

( M ) ( L ) = ( M ) x ( L )

( T )2 ( T )2

( M ) ( L ) = ( M ) ( L )( T )2 ( T )2

LEFT = RIGHT


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Dimensional Analysis

This is a very important tool to check your workEg. : Doing a problem you get the answer distance

d = v t2 (velocity x time2)

Units on left side = ( L )

Units on right side = ( L )/( T ) x ( T )2 = ( L ) .( T )

Left units and right units dont match, soLeft units and right units dont match, so

answer must beanswer must be wrong!!wrong!!


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Exercise

The periodThe period PP of a swinging pendulumof a swinging pendulumdepends only on the length of the pendulumdepends only on the length of the pendulumdd and the acceleration of gravityand the acceleration of gravity gg..

Which of the following formulas forWhich of the following formulas forPP

couldcould be correct ?be correct ?

g

dP T2!(a)(a) P = 2T(dg)2 (b)(b) (c)(c) P

d

g! 2

Given : d has units of length ( L ) and

g has units of ( L / T 2).


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Realize that the left hand side P

has units of time (TT ) Try the first equation

P dg(a)(a) (b)(b) (c)(c)

(a)(a) LL

T

L

TT

{

2

2 4

4Not Right !!Not Right !!

Pd

g! 2TP

d

g! 2T


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L

L

T

T T

2

2! {

P dg!22

T(a)(a) (b)(b) (c)(c)

(b)(b) Not Right !!Not Right !!

Try the second equation

Pd

g! 2TP

d

g!2T


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TT

T

L

L 2

2

!!

P dg! T(a)(a) (b)(b) (c)(c)

(c)(c) This has the correct units!!This has the correct units!!

This must be the answer!!This must be the answer!!

Try the third equation

Pd

g! 2TP

d

g! 2T


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If an equation is dimensionally homogeneous

butits additive terms have inconsistent unit, the

terms may be made consistent by applying

conversion factors

Example:

V (m/s) = Vo (m/s) + g (m/s2) t (min)

Apply the conversion factor

V (m/s) = Vo (m/s) + g (m/s2) t (min)(60s/min)

V= Vo + 60 g tV= Vo + 60 g t


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An equation is onlyVALIDwhen it is dimensionally

Homogeneous &

consistent in UNITS!!!


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Dimensionless Quantities

Can be a pure number

Eg. : 2, 1.3 ,5/2

a multiplicative combination of variables with

no net dimensions

Eg. :

Q

Vud!Re

= (g/cm= (g/cm33) , u = (cm/s),) , u = (cm/s),d = (cm), = (g/cm.s)d = (cm), = (g/cm.s)

DIMENSIONLESS


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DIMENSIONLESS GROUPS

V DR

V

Q

V = density (kg/m3) [M/L3]

V= velocity (m/s) [L/T]

D = diameter (m) [L]

Q = viscosity (kg/m.s) [M/L.T]R = [M/L3]*[L/T]*[L]*[L.T/M]

Ratio of inertial to viscous forces used in fluid flow

Small values of R laminar flow

Large values of R turbulent flow


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So far.Unitsdimensions

Conversion of unitsMass, moleT, P

All important concepts and techniques forstudying systems and developing andsolving material balances

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Law Chapter 1 Part2 2011