# 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