Post on 12-May-2021
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