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LIVETUTORIALS LEARNING LIBRARY

Dimensional Consistency and Unit Conversion

Distinguish units from dimensions, combine dimensional powers, and reject equations whose terms represent incompatible quantities.

By LiveTutorials

Overview

Write the dimensions of familiar quantities before testing an unfamiliar formula. Displacement has dimension L, velocity has LT^-1, and acceleration has LT^-2. A numerical constant without units is dimensionless and does not change the dimensional powers of the quantity it multiplies.

Check a sum term by term. If one term has dimension L and another LT^-1, they cannot be added as two contributions to the same displacement. For products, add powers; for quotients, subtract powers. Thus acceleration multiplied by time squared has dimension LT^-2 × T^2 = L.

Unit conversions preserve the quantity by multiplying by ratios equal to one. To convert kilometres per hour to metres per second, multiply by 1,000 metres per kilometre and divide by 3,600 seconds per hour. Keep the units written out so the cancellation shows the correct direction.

The tutorial below combines symbolic checks with a numerical example. Its final check asks what dimensional analysis cannot tell you, which is as important as recognising what it can detect.

References

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