Dimensional Analysis

This lesson covers dimensional analysis — a powerful technique for checking equations, deriving possible forms of relationships, and converting between unit systems.

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1 Fundamental Dimensions

All mechanical quantities can be expressed in terms of three fundamental dimensions:

Dimension	Symbol
Mass	$M$M
Length	$L$L
Time	$T$T

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2 Dimensions of Common Quantities

Quantity	Formula	Dimensions
Displacement	$s$s	$L$L
Velocity	$s/t$s/t	$LT^{-1}$LT−1
Acceleration	$v/t$v/t	$LT^{-2}$LT−2
Force	$ma$ma	$MLT^{-2}$MLT−2
Momentum	$mv$mv	$MLT^{-1}$MLT−1
Energy / Work	$Fs$Fs	$ML^2T^{-2}$ML2T−2
Power	$W/t$W/t	$ML^2T^{-3}$ML2T−3
Pressure	$F/A$F/A	$ML^{-1}T^{-2}$ML−1T−2
Angular velocity	$\theta/t$θ/t	$T^{-1}$T−1
Moment of inertia	$mr^2$mr2	$ML^2$ML2
Torque	$Fr$Fr	$ML^2T^{-2}$ML2T−2
Spring constant	$F/x$F/x	$MT^{-2}$MT−2
Gravitational const $G$G	$Fr^2/(m_1 m_2)$Fr2/(m1​m2​)	$M^{-1}L^3T^{-2}$M−1L3T−2

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3 Principle of Dimensional Homogeneity

A physically valid equation must be dimensionally homogeneous: every term must have the same dimensions.