LIVETUTORIALS LEARNING LIBRARY
Units, Measurements and Dimensional Reasoning
Keep units through a calculation and compare the dimensions of every term in a physical equation.
Overview
A measured quantity is expressed as a numerical value multiplied by a unit. When the unit changes, the numerical value must change in the opposite way so that the quantity remains the same. For example, one kilometre is one thousand metres; a distance expressed in kilometres therefore has a smaller numerical value than the same distance expressed in metres.
Derived quantities combine base quantities. Speed is distance divided by time, acceleration is change in velocity divided by time, and force in Newtonian mechanics is mass multiplied by acceleration. Their dimensions can be written using M for mass, L for length and T for time.
In an equation containing a sum, each term must represent the same dimensional kind of quantity. A length can be added to another length, but adding a length directly to a speed is not meaningful. Products and quotients combine dimensions using the same exponent rules used in algebra.
A consistent dimensional result is a screening test rather than a proof. Dimensionless factors such as one half remain invisible to it, and different physical quantities can share dimensions. Use the check alongside the stated assumptions, numerical reasoning and physical interpretation.
References
Continue learning
- Dimensional Consistency and Unit Conversion
Distinguish units from dimensions, combine dimensional powers, and reject equations whose terms represent incompatible quantities.