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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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