Assume a Product of Powers and Solve for the Exponents
When a question says a quantity depends on a few others, write it as a product of powers with an unknown constant and match the dimensions of mass, length and time. Each base dimension gives one equation.
Deriving Relations by Dimensional Analysis is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
The method
For the critical velocity of a liquid in a tube, v_c = k η^x ρ^y r^z gives [L T⁻¹] = [M L⁻¹ T⁻¹]^x [M L⁻³]^y [L]^z, so x + y = 0, -x - 3y + z = 1 and -x = -1, hence x = 1, y = -1, z = -1. For the period of a vibrating drop depending on surface tension S, density ρ and radius r, T = k S^a r^b ρ^c gives a = -1/2, c = 1/2 and b = 3/2, so T = k sqrt(ρ r³ / S).
Building a length from constants
Dimensional analysis cannot give dimensionless constants such as 2π or 1/2, and it cannot handle a sum of terms or functions such as sine.
OlympIQ exam lens
- ✓Write Q = k x^a y^b z^c and equate powers of M, L, T
- ✓Critical velocity: v_c is proportional to η/(ρ r)
- ✓Drop oscillation: T = k sqrt(ρ r³ / S)
- ✓Dimensional analysis misses pure numbers and cannot treat sums or trigonometric functions
Do not assume the unknown constant k is 1. The method fixes the powers of the variables only; the numerical factor needs experiment or theory.
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