1 INTRODUCTION 1.7 Dimensional analysis
×3 zWorked example 1.2: Tire pressure
Question: The recommended tire pressure in a Honda Civic is 28 psi (pounds per
square inch). What is this pressure in atmospheres (1 atmosphere is 105 N m−^2 )?
Answer: First, 28 pounds per square inch is the same as 28 (12)^2 = 4032 pounds
per square foot (the standard fps unit of pressure). Now, 1 pound equals 4.
Newtons (the standard SI unit of force), and 1 foot equals 0.3048 m (see Tab. 2 ).
Hence,
P = 4032 × (4.448)/(0.3048)^2 = 1.93 × 105 Nm−^2.It follows that 28 psi is equivalent to 1.93 atmospheres.
Worked example 1.3: Dimensional analysis
Question: The speed of sound v in a gas might plausibly depend on the pressure p,
the density ρ, and the volume V of the gas. Use dimensional analysis to determine
the exponents x, y, and z in the formula
v = C px ρy Vz,where C is a dimensionless constant. Incidentally, the mks units of pressure are
kilograms per meter per second squared.
Answer: Equating the dimensions of both sides of the above equation, we ob-
tain
[L] [M]
x
[M]
y
[T ]=
[T 2 ][L]
[L^3 ][L ].A comparison of the exponents of [L], [M], and [T ] on either side of the above
expression yields
1 = −x − 3y + 3z,
0 = x + y,
− 1 = −2x.