B.1

Mechanics lecture 2 Applications of dimensional analysis

Dr Philip Jackson

http://www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3/

EE1.el3 (EEE1023): Electronics III

B.2 Applications of dimensional analysis

•  Constants –  Definition of a physical constant –  Examples of constants, values, units and dimensions

•  Dimensions of formulae

•  Dimensionless quantities –  Motivation –  Popular measures –  Use for scaling

B.3 Constants Mathematical constants

Physical constants

Archimedes’ constant π 3.14159… Euler’s number e 2.71828…

speed of light c 3.00×108 ms-1 gravitation G 6.67×10-11 m3kg-1s-2

gravity g 9.81 ms-2

Plank’s constant h 6.63×10-34 Js

elementary charge e 1.60×10-19 C electron mass me 9.11×10-31 kg electric permittivity ε0 8.85×10-12 Fm-1

magnetic permeability µ0 4π×10-7 mNA-2

Avagadro’s number NA 6.02×1023 mol-1 gas constant R 8.31 JK-1mol-1 atmosphere atm 1.01×105 Pa reference sound pressure pref 2.00×10-5 Pa

source: Wikipedia

source: NASA

B.4 Dimensions in equations

Quantity Dim. Unit Symbol

mass M kilogram kg

length L metre m

time T second s

current I ampere A

temperature Θ kelvin K

substance N mole mol

luminosity J candela cd

B.5 Checking an equation for transient voltage

V(t)

time, t

V(t) = V0 e-t/RC

Is this plausible?

The exponent t/RC must be a simple number – dimensionless

Resistance R (Ohms), and capacitance C (Farads)

R = V/I, C =Q/V, so RC = Q/I dimensions of charge

dimensions of current dim(RC) = T

So, dim(t/RC) = T T-1, which is dimensionless - OK!

B.6

The dimensions on both sides of an equation must agree.

Example:

For a fluid in motion, pressure depends on density ρ and velocity V. What combination of ρ and V gives the right dimensions of pressure?

P = k ρx Vy Question: what are x and y?

Pressure P has dimensions ML-1T-2 (force per unit area ((MLT-2)/(L2)): dim(P) = ML-1T-2, dim(ρ) = ML-3 and dim(V) = LT-1

P / ρ = (ML-1T-2)M-1L3 = L2T-2, which has the dimensions of V2

So, x = 1 and y = 2.

This leads us towards an equation of the form P = k ρ V2

which compares well to Bernoulli’s equation, P = ½ ρ V2

Pressure in moving fluid

B.7 Dimensionless quantities

•  Ratios

•  Fluids

•  Materials

•  Electrical

Dimensionless quantities allow comparison, generalisation and scaling of experimental results.

B.8 Ratio examples

•  Fraction 1/4

•  Ratio 1:3

•  Percentage 50%

•  Angle π/4 radians

•  Shape factor area / (max. length × max. width)

B.9 Fluid examples

•  Mach number

where c0 is speed of sound

•  Reynolds number

where µ0 is dynamic viscosity of fluid

•  Lift coefficient

where L is lift force

!

M =V

c0

!

Re ="VD

µ

!

CL

=L

1

2"V 2

A

B.10 Material examples

•  Refractive index

where vP is the phase velocity in the medium

•  Static friction coefficient

where F is the friction force and N the normal force

•  Strain

!

n =c

vP

!

µS

=F

N

!

" =#L

L

B.11 Electrical examples

•  Gain

•  Decibel e.g., 3 dB

•  Power factor

!

A =Vout

Vin

!

10log10 A

factor =real _ power

apparent _ power

B.12 Dimensionless quantities

•  Ratios –  ratio, percentage, radian

•  Fluids –  Mach number, Reynolds number, drag & lift coefficients

•  Materials –  refraction index, friction coefficient, Poisson ratio, strain

•  Electrical –  gain, decibel, power factor

To compare, generalise and scale experimental results.

B.13 Generalisation through scaling

•  Forces in wind tunnel testing

•  Acceleration of a mag-lev train

•  Interference of waves

B.14

How long would it take to boil an ostrich egg?

Assume round eggs, chicken egg rch = 3 cm, ostrich ros =15 cm, and that it takes 3 min to boil a chicken egg.

Factors: specific heat capacity c, thermal conductivity k, density ρ, time t, radius r.

c has units J/kg K = kg m s-2 m/kg K = m2/s2 K

k has units J/s m K = kg m s-2 m/s m K = kg m/s3 K

r has units m

t has units s

ρ has units kg/m3

We can find a dimensionless group, D = kt/ρcr2

tos/tch = ros2/rch

2 = 225/9 = 25, so it takes 25 x 3 min = 75 min.

Scaling to boil an ostrich egg

B.15

Using the same dimensionless group, D = kt/ρcr2, how long would it take to cook the golden egg from the golden goose?

Hint: density of gold, ρAu = 19000 kg/m3 (ρch = 1000), kAu = 304 J/smK (kch = 1), cAu = 120 J/kgK (cch = 4200)

The key is that the dimensionless quantity remains unchanged:

Dgg = Dch

Eggsample from the golden goose

tgg = (ρgg/ρch) (cAu/cch) (rgg2/rch

2) (kch/kAu) tch

= (19000/1000) x (120/4200) x (82/32) x (1/ 304) x 7min

= 19 x 0.029 x 7.1 x 0.003 x 7 x 60 s

= 5 s.

B.16 Summary of dimensionless units

•  Constants –  mathematical constants –  physical constants

•  Dimensions of formulae

•  Dimensionless quantities –  ratios and angles –  fluid, material and electrical

•  Scaling –  Applications and examples

B.17 Force vectors in equilibrium

•  Forces –  magnitude and direction –  components of a force

•  Equilibrium –  static or dynamic –  forces in balance

•  Preparation –  What are the fundamental forces? List them –  What is a component of force? Give one example –  What is equilibrium? Draw a diagram with 3-4 forces