Chapter1 Short Notes Physics

Dimensional Analysis Cheat Sheet

Quick Rules & Cheat Codes

• Dimensionless: Relative Density, Angular Displacement ($\theta =\frac{\text{arc}}{\text{radius}}=[M^0{L}^0{T}^0]$)
• “Gradient” Trick: Divide PQ by $[L]$. (e.g., Temp Gradient = $[K{L}^{-1}]$, Velocity Gradient = $[T^{-1}]$)
• “Density” Trick: Divide PQ by $[L^3]$ (Volume). (e.g., Energy Density = $\frac{[M{L}^2{T}^{-2}]}{[L^3]}=[M{L}^{-1}{T}^{-2}]$)
• “Time-Alikes” $[T]$: $RC$ (Res $\times$ Cap), $L/R$ (Inductance / Res), $\frac{1}{\text{Frequency}}$

Important Examples

Physical Quantity	Original Formula / Relation	Unit	Dimension
Velocity / Speed	$\frac{\text{disp}}{\text{time}},\frac{\text{dist}}{\text{time}}$	$m{s}^{-1}$	$[L{T}^{-1}]$
Acceleration	$\frac{\text{velocity}}{\text{time}}$	$m{s}^{-2}$	$[L{T}^{-2}]$
Force	$m\cdot a$	$kgm{s}^{-2}$	$[ML{T}^{-2}]$
Pressure	$\frac{\text{force}}{\text{area}}$	$N{m}^{-2}$	$[M{L}^{-1}{T}^{-2}]$
Momentum	$P=mv$	$kgm{s}^{-1}$	$[ML{T}^{-1}]$
Work	$F\cdot \text{disp}$	$kg{m}^2{s}^{-2}$	$[M{L}^2{T}^{-2}]$
Power	$\frac{\text{work}}{\text{time}}$	$kg{m}^2{s}^{-3}$	$[M{L}^2{T}^{-3}]$
Intensity	$\frac{\text{Power}}{\text{Area}}$	$\text{W}/{\text{m}}^2$	$[M{T}^{-3}]$
Density	$\frac{\text{mass}}{\text{volume}}$	$kg{m}^{-3}$	$[M{L}^{-3}]$
Momentum / Impulse	$Force\cdot Time$	$Newton\cdot s$	$[ML{T}^{-1}]$
Angular Momentum	$\text{Distance}\times \text{Momentum}$	-	$[M{L}^2{T}^{-1}]$
Moment of Inertia	$\text{Mass}\times {\text{Radius}}^2$	-	$[M{L}^2]$

Tricky & Advance Dimensions

Physical Quantity	Original Formula / Relation	Unit	Dimension
Charge	$I=\frac{q}{t}$	$\text{culomb(C)}$	$[AT]$
Resistance	$P=I^{2}R$	$\Omega$	$[M{L}^2{T}^{-3}{A}^{-2}]$
Voltage/Potential	$P=VI$	$volt(V)$	$[M{L}^2{T}^{-3}{A}^{-1}]$
Electric Field	$F=qE$	$V{m}^{-1}/N{C}^{-1}$	$[ML{T}^{-3}{A}^{-1}]$
Magnetic Field	$E=Bc$	$\text{Tesla}(T)$	$[M{T}^{-2}{A}^{-1}]$
Gravitational Constant ($G$)	$F=\frac{G{m}_1{m}_2}{r^2}$	${\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$	$[M^{-1}{L}^3{T}^{-2}]$
Boltzmann Constant ($k_B$)	$E=k_B\times \text{Temperature}$	$\text{J/K}$	$[M{L}^2{T}^{-2}{K}^{-1}]$
Planck’s Constant ($h$)	$E=hf$	$\text{J}\cdot \text{s}$	$[M{L}^2{T}^{-1}]$
Coefficient of Viscosity ($\eta$)	$F=\eta rv$	$\text{Pa}\cdot \text{s}$	$[M{L}^{-1}{T}^{-1}]$
Permittivity of Free Space (${ϵ}_0$)	$F=\frac{1}{4\pi {ϵ}_0}\frac{q^2}{r^2}$	-	$[M^{-1}{L}^{-3}{T}^4{A}^2]$
Permeability of Free Space (${\mu }_0$)	$c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$	-	$[ML{T}^{-2}{A}^{-2}]$
Speed Relationship	$\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$	$c$	$[L{T}^{-1}]$
Rydberg Constant ($R$)			$[L^{-1}]$

List of PQs with same Dimensions

Category	Dimension	Physical Quantities Sharing This Dimension	Exceptions / Notes
Distance	$[L]$	Distance, Displacement, Radius of Gyration, Light Year, Parsec, Astronomical Unit (AU)	Angular Displacement = $[M^0{L}^0{T}^0]$
Speed	$[L{T}^{-1}]$	Speed, Velocity, Average Speed, Average Velocity, Terminal Velocity, Critical Velocity, Velocity of Light, Escape Velocity, Orbital Velocity, Relative Velocity, Instantaneous Velocity	Angular Velocity = $[T^{-1}]$
Acceleration	$[L{T}^{-2}]$	Acceleration, Average Acceleration, Instantaneous Acceleration, Acceleration due to Gravity, Intensity of Gravitational Field, Centripetal Acceleration, Centrifugal Acceleration	Angular Acceleration = $[T^{-2}]$
Force	$[ML{T}^{-2}]$	Force, Centripetal Force, Centrifugal Force, Friction, Gravitational Force, Thrust, Viscous Force, Spring Force, Magnetic Force, Non-conservative Force, Radiation Force, Tension, Normal Reaction, Weight, Restoring Force, Electrostatic Force, Lorentz Force, Buoyant Force	Surface Tension & Spring Constant = $[M{T}^{-2}]$
Energy	$[M{L}^2{T}^{-2}]$	Work, Energy, Kinetic Energy, Potential Energy, Heat Energy, Thermal Energy, Vibrational Energy, Moment of Force, Torque or Couple, Strain Energy
Pressure	$[M{L}^{-1}{T}^{-2}]$	Pressure, Stress, Young Modulus, Bulk Modulus, Shear Modulus, Modulus of Rigidity, Energy Density