AcademyMeasurement and Vectors
Academy
Units and Measurement Standards
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Units attach numbers to reproducible measurement standards.
Notation
\(L\)
length dimension
\(\mathrm{m}\)
\(M\)
mass dimension
\(\mathrm{kg}\)
\(T\)
time dimension
\(\mathrm{s}\)
\([q]\)
dimension of quantity q
varies
\(N\)
newton
\(\mathrm{kg\,m\,s^{-2}}\)
\(J\)
joule
\(\mathrm{kg\,m^{2}\,s^{-2}}\)
Method
Derived units are built by replacing each physical quantity with its base dimensions.
Speed
\[[v]=[x/t]=LT^{-1}\]
Acceleration
\[[a]=[v/t]=LT^{-2}\]
Force
\[[F]=[ma]=M(LT^{-2})\]
Energy
\[[E]=[Fx]=(MLT^{-2})L\]
Rules
Speed dimension
\[[v]=LT^{-1}\]
Acceleration dimension
\[[a]=LT^{-2}\]
Force dimension
\[[F]=MLT^{-2}\]
Energy dimension
\[[E]=ML^2T^{-2}\]
Checks
- Added terms need identical dimensions.
- Units can change while dimensions stay fixed.
- Derived units must reduce to base units.
- A dimensionless ratio has no remaining unit.