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CHAPTER 1.3: DIMENSIONS CHAPTER 1.3 DIMENSIONAL ANALYSIS

Chapter 1(3)DIMENSIONAL ANALYSIS

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Page 1: Chapter 1(3)DIMENSIONAL ANALYSIS

CHAPTER 1.3: DIMENSIONS

CHAPTER 1.3

DIMENSIONAL ANALYSIS

Page 2: Chapter 1(3)DIMENSIONAL ANALYSIS

Grab the whole picture !

Measurements Quantities

Units Instruments

Vector QuantitiesScalar Quantities

Accuracy & UncertaintyDimension Analysis Significant Figures

DIMENSIONAL ANALYSIS

Page 3: Chapter 1(3)DIMENSIONAL ANALYSIS

DIMENSIONSDIMENSIONS

What is “Dimension” ?

Page 4: Chapter 1(3)DIMENSIONAL ANALYSIS

Many physical quantities can be expressed in terms of aMany physical quantities can be expressed in terms of a combination ofcombination of fundamental dimensions such asfundamental dimensions such as

[Length] L[Time] T[Mass] M[Current] A[Temperature] θ[Amount] N

The symbol [ ] means dimension or stands for dimensionThe symbol [ ] means dimension or stands for dimension

Page 5: Chapter 1(3)DIMENSIONAL ANALYSIS

There are physical quantities which are dimensionless:

numerical value ratio between the same quantity angle some of the known constants like ln,

log and etc.

Page 6: Chapter 1(3)DIMENSIONAL ANALYSIS

Dimensional AnalysisDimensional Analysis

Dimension analysis can be used to:

Derive an equation.

Check whether an equation is dimensionally correct. However, dimensionally correct doesn’t necessarily mean the equation is correct

Find out dimension or units of derived quantities.

Page 7: Chapter 1(3)DIMENSIONAL ANALYSIS

Derived an Equation (Quantities)Derived an Equation (Quantities)

Example 1

Velocity = displacement / time

[velocity] = [displacement] / [time]

= L / T

= LT-1

v = s / t

Page 8: Chapter 1(3)DIMENSIONAL ANALYSIS

Example 2 The period of a pendulum

The periodThe period PP of a swinging pendulum depends only on the length of of a swinging pendulum depends only on the length of the pendulumthe pendulum ll and the acceleration of gravityand the acceleration of gravity gg..

What are the dimensions of the variables?What are the dimensions of the variables?

t → T m → M ℓ → L g → LT-2

Page 9: Chapter 1(3)DIMENSIONAL ANALYSIS

T = kma ℓbgc

Write a general equation:Write a general equation:

T α ma ℓ bgc

By using the dimension method, an expression could By using the dimension method, an expression could be derived that relates T, l and gbe derived that relates T, l and g

whereby a, b and c are dimensionless constantwhereby a, b and c are dimensionless constant

thusthus

Page 10: Chapter 1(3)DIMENSIONAL ANALYSIS

Write out the dimensions of the variablesWrite out the dimensions of the variables

T = MaLb(LT-2)c

T = MaLbLcT-2c

T1 = MaLb+cT-2c

[T] = [ma][ℓ b][gc]

Using indicesUsing indices

a = 0 -2c = 1 → c = -½

b + c = 0 b = -c = ½

Page 11: Chapter 1(3)DIMENSIONAL ANALYSIS

T = kma ℓbgc

T = km0 ℓ½g-½

g

lkT

Whereby, the value of k is known by experimentWhereby, the value of k is known by experiment

Page 12: Chapter 1(3)DIMENSIONAL ANALYSIS

ExercisesExercises

The viscosity force, F going against the movement of a sphere immersed in a fluid depends on the radius of the sphere, a the speed of the sphere, v and the viscosity of the fluid, η. By using the dimension method, derive an equation that relates F with a, v and η.

(given that )Av

Fl

Page 13: Chapter 1(3)DIMENSIONAL ANALYSIS

To check whether a specific formula or To check whether a specific formula or an equation is homogenousan equation is homogenous

Example 1

S = vt

[s] = [v] [t]

L.H.S[s] = L

R.H.S

[v] [t] = LT-1(T)

[v] [t] = LThus, the left hand side = right hand side, rendering the equation as homogenous

Page 14: Chapter 1(3)DIMENSIONAL ANALYSIS

m

FC

][

][][ 2

m

FC

Example 2

Given that the speed for the wave of a rope is ,

Check its homogenity by using the dimensional analysis

m

FC 2

Page 15: Chapter 1(3)DIMENSIONAL ANALYSIS

L.H.S

[C] = (LT-1)2

[C] = L2T-2

R.H.S

[F] = MLT-2 ,

= LT-2

M

MLT

M

F 2

][

][

[M] = M

Conclusion: The above equation is not homogenous

(L.H.S ≠ R.H.S)

Page 16: Chapter 1(3)DIMENSIONAL ANALYSIS

ExercisesExercises

Show that the equations below are

either homogenous or otherwise

v = u + 2as

s = ut + ½ at2

Page 17: Chapter 1(3)DIMENSIONAL ANALYSIS

Find out dimension or units of derived Find out dimension or units of derived quantitiesquantities

k

mT 22

k

mT 2

Example

Consider the equation ,

where m is the mass and T is a time, therefore dimension of k can be describe as

k

mT 2

Page 18: Chapter 1(3)DIMENSIONAL ANALYSIS

2

2

T

mk

][

][][

2T

mk

2T

M

2MT → unit: kgs-2

thus, the units of k is in kgs-2

Page 19: Chapter 1(3)DIMENSIONAL ANALYSIS

ExerciseExercise

The speed of a sound wave, v going through an elastic matter depends on the density of the elastic matter, ρ and a constant E given as equation

V = E½ - ρ-½

Determine the dimension for E in its SI units

Page 20: Chapter 1(3)DIMENSIONAL ANALYSIS

SCALAR AND VECTOR

Page 21: Chapter 1(3)DIMENSIONAL ANALYSIS

Dimensional AnalysisDimensional Analysis

Example:Example:

The periodThe period PP of a swinging pendulum depends only on of a swinging pendulum depends only on the length of the pendulumthe length of the pendulum ll and the acceleration of and the acceleration of gravitygravity gg.. Which of the following formulas forWhich of the following formulas for PP couldcould be correct ?be correct ?

g

lP 2

g

lP 2(a)(a) (b)(b) (c)(c)

Given: d has units of length (L) and g has units of (L / T 2).

P = 2 (lg)2

Page 22: Chapter 1(3)DIMENSIONAL ANALYSIS

Dimensional AnalysisDimensional Analysis

Example continue…Example continue…Realize that the left hand side P has units of time (TT )

Try the first equation

P dg2 2(a)(a) (b)(b) (c)(c)

(a)(a) LL

T

L

TT

2

2 4

4 Not Right !!Not Right !!

Pdg

2Pdg

2

Page 23: Chapter 1(3)DIMENSIONAL ANALYSIS

LL

T

T T

2

2

P dg2 2(a)(a) (b)(b) (c)(c)

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

Example continue…Example continue…Try the second equation

Dimensional AnalysisDimensional Analysis

Pdg

2Pdg

2

Page 24: Chapter 1(3)DIMENSIONAL ANALYSIS

TT

TLL 2

2

P dg2 2(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!!

Example continue…Example continue…

Try the third equation

Dimensional AnalysisDimensional Analysis

Pdg

2Pdg

2