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Dimensional Analysis:

Most physical quantities can be expressed in terms of five basic dimensions, listed below with their SI units (see the bottom of this page):

  • Mass, M, kg
  • Length, L, m
  • Time, T, s
  • Electric Current, I, A
  • Temperature, θ, K

Only quantities with like dimensions may be added, subtracted or compared. This rule provides a powerful tool for checking whether equations are dimensionally consistent. It is also possible to use dimensional analysis to suggest plausible equations when we know which quantities are involved.

 

Example of checking for dimensional consistency:

Consider one of the equations of constant acceleration,

s = ut + 1/2 at2. (1)

The equation contains three terms: s, ut and 1/2at2. All three terms must have the same dimensions.

  • s: displacement = L
  • ut: velocity x time = LT-1 x T = L
  • 1/2at2 = acceleration x time squared = LT-2 x T2 = L

All three terms have units of length and hence this equation is dimensionally valid. Of course, this does not tell us if the equation is physically correct, nor does it tell us whether the constant 1/2 is correct or not!

 

Example of generating equations

Let’s say that you drop a ball out of a window on Earth and want to know the final velocity of the ball as it hits the ground. We can solve this problem using dimensional analysis!

First consider the dimensions of the velocity and the other factors that may affect it:

  • Final velocity, v: LT-1
  • Mass, m: M
  • Distance fallen, s: L
  • The acceleration, a: LT-2

Our equation is going to take the form v = Cmαsβaγ where the power constants α, β and γ are unknown and C is some dimensionless constant.

 

Re-writing our equation using dimensions: LT-1 = (M)α (L)β (L T-2)γ

To be dimensionally consistent, each dimension must appear to the same power on each side. Hence:

  • For L: 1 = β + γ
  • For M: 0 = α so mass is not present in the equation.
  • For T: -1 = -2γ

Solving these equations, we get: α=0, β=1/2 and γ= 1/2.

Hence, v = Cs1/2a1/2 where C is an arbitrary constant. This should be recognisable to you as an equation of motion (v2 = u + 2as with u = 0ms-1 as we are starting from stationary) that you’ve seen before!

Constants and SI Units:

All SI units written in the form described by NIST.

Table 1: SI Base Units.

Dimensions

Name

Symbol

Length

metre

m

Mass

kilogram

kg

Time

second

s

Electric Current

ampere

A

Thermodynamic Temperature

kelvin

K

Amount of Substance

mole

mol

Luminous Intensity

candela

cd

 

 

Table 2: SI Derived Units

Dimension

Name

Symbol

Area

square metre

m2

Volume

cubic metre

m3

Speed/Velocity

metre per second

m s-1

Acceleration

metre per second squared

m s-2

Wave Number

reciprocal metre

m-1

Density/Mass Density

kilogram per cubic metre

kg m-3

Specific Volume

cubic metre per kilogram

m3kg-1

Current Density

ampere per square metre

A m-2

Magnetic Field Strength

ampere per metre

A m-1

Concentration

mole per cubic metre

mol m-3

Luminance

candela per square metre

cd m-2

 

Table 3: SI Derived Units with Special Names

Dimension

Name

Symbol

Standard Units

Base Units

Frequency

hertz

Hz

 

s-1

Force

newton

N

 

kg m s-2

Pressure/Stress

pascal

Pa

N m-2

kg m-1 s-2

Energy/Work/Quantity of Heat

joule

J

N m

kg m2 s-2

Power

watt

W

J s-1

kg m2 s-3

Electric Charge

coulomb

C

 

A s

Voltage

volt

V

W A-1

kg m2 s-3 A-1

Capacitance

farad

F

C V-1

kg-1 m-2 s4 A2

Inductance

henry

H

 

A-2 kg m2 s-2

Electric Resistance

ohm

 

V A-1

kg m2 s-3 A-2

Celsius Temperature

degree Celsius

C

 

K

Activity of a Radionuclide

becquerel

Bq

 

s-1

Dynamic Viscosity

pascal second

 

Pa s

kg m-1 s-1

Moment of Force

newton meter

 

N m

kg 

Heat Capacity/Entropy

joule per kelvin

 

J K-1

kg m2 s-2 K-1 

Specific Heat Capacity/Specific Entropy

joule per kilogram kelvin

 

J kg-1 K-1 

m2 s-2 K-1 

Specific Energy

joule per kilogram

 

J kg-1 

m2 s-2 

Thermal Conductivity

watt per metre kelvin

 

W m-1 K-1 

kg m s-3 K-1 

Energy Density

joule per cubic metre

 

J m-3 

kg m-1 s-2 

Electric Field Strength

volt per metre

 

V m-1 

kg m s-3 A-1 

 

Table 4: A selection of physical constants from CODATA 2018

Name

Symbol

Value (standard uncertainty in brackets)

Standard acceleration of gravity (exact)

g

9.806 65 ms2

Unified atomic mass unit

u

1.660 539 066 60(50) x 10-27 kg 

Avogadro constant (exact)

NA

6.022 140 76 x 1023 mol-1   

Bohr magneton

μB

9.274 010 0783(28) x 10-24 J T-1   

Bohr radius

ao

5.291 772 109 03(80) x 10-11 m   

Boltzmann constant (exact)

k

1.380 649 x 10-23 J K-1   

Elementary charge (exact)

e

1.602 176 634 x 10-19 C   

Electron mass

me

9.109 383 7015(28) x 10-31 kg   

Fine structure constant

α

7.297 352 5693(11) x 10-3    

Inverse fine structure constant

1/α

137.035 999 084(21) 

Newtonian constant of gravitation

G

6.674 30(15) x 10-11 m3 kg-1 s-2    

Nuclear magneton

μN

5.050 783 7461(15) x 10-27 J T-1

Vacuum magnetic permeability

μ0

1.256 637 062 12(19) x 10-6 N A-2

Vacuum electric permittivity

ε0

8.854 187 8128(13) x 10-12 F m-1  

pi

π

3.141 592 653...

Planck constant (exact)

h

6.626 070 15 x 10-34 J Hz-1 

Proton mass

mp

1.672 621 923 69(51) x 10-27 kg   

Speed of light in vacuum (exact)

c

299 792 458 ms-1

Stefan-Boltzmann constant (exact)

σ

5.670 374 419... x 10-8 W m-2 K-4