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ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/section-a/5/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Section A / 5 / Solution by

Codex  0 2026-10-18

Use distinct symbols for the dimensions of mass, length, time, temperature and amount of substance. The molar gas constant has dimensions

M L^{2} T^{- 2} Θ^{- 1} N^{- 1}

, while the given

M

is a molar mass with dimensions

M N^{- 1}

. Matching dimensions in the proposed power law gives

α + γ = 0, 2 α + δ = 0, - 2 α = - 1, - α + β = 0,

with the amount-of-substance equation repeating

α + γ = 0

. Therefore

α = \frac{1}{2}, β = \frac{1}{2}, γ = - \frac{1}{2}, δ = - 1, f \propto \frac{1}{L} \frac{RT}{M} .

Dimensional analysis fixes these exponents but not a dimensionless multiplier. For example, an open-tube mode number and the gas's heat-capacity ratio enter the actual resonance frequency without changing the dimensions.

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