Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version 8th Edition

Published by Wiley
ISBN 10: 1119080703
ISBN 13: 978-1-11908-070-1

Chapter 1 - Problems: 1.2

Answer

The Mach number, Ma, is dimensionless.

Work Step by Step

For an airplane flying at velocity V in air at absolute temperature T, the Mach number is: $Ma=\frac{V}{\sqrt kRT}$ Where (Using the International System of Units), $V=\frac{m}{s}=LT^{-1}$ $K=dimensionless$ (Given) $R=\frac{J}{KgK}=\frac{Nm}{KgK}=\frac{Kgm^{2}}{s^{2}KgK}=\frac{m^{2}}{s^{2}K}=L^{2}T^{-2} Θ^{-1}$ $T=K= Θ $ Therefore, $Ma=LT^{-1}(L^{2}T^{-2}Θ^{-1}\timesΘ)^{-1/2}=LT^{-1}(L^{2}T^{-2}Θ^{0})^{-1/2}=LT^{-1}(LT^{-1}Θ^{0})^{-1}$ $Ma=L^{0}T^{0}Θ^{0}$ Thus, it's proven that the Mach number, Ma, is dimensionless.
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