You can add two lengths, but you cannot add a length to a time. This simple rule is very useful. In this course we write the dimensions of a physical quantity, we use them to check an equation, we use them to find a formula that we do not remember, and we also learn what dimensions cannot tell us.
3 lessons, about 1 h 15 min3 simulations12 problems6 practice questionsPython in every lesson
Created by StemPy LabThe dimensions of a quantity are the powers of the base quantities that make it, written in square brackets: a speed is , a force , an energy . Find them from a formula, multiplying and dividing as the formula does. Some quantities have no dimensions (angle, strain, refractive index), and different quantities can share the same dimensions (work and torque).
Only quantities of the same kind can be added, subtracted or equated, so every term of a correct equation has the same dimensions (the principle of homogeneity). That lets us catch wrong equations and find the dimensions of a constant. The argument of sin, cos, log or e must be a pure number. A check can prove an equation wrong, never right: it cannot see a pure number like 2 or a missing term of the right kind.
If we know which quantities a result depends on, dimensions can give the formula up to a pure number: write it as a product of powers, , and match the powers of M, L and T. Dimensions also change a value from one system of units to another: . The method cannot find a number like , cannot give a sum of terms, and fails when more than three quantities are involved in mechanics.
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