The speed v of a longitudinal wave propagating through a gas depends on the pressure P of … — JAMB UTME

The speed $v$ of a longitudinal wave propagating through a gas depends on the pressure $P$ of the gas and its density $\rho$ according to the dimensional relationship $v = C P^x \rho^y$ , where $C$ is a dimensionless constant. Using dimensional analysis, what is the numerical value of $x - y$ ?

Answer: 1

Answer

The numerical value of

x - y

is 1.0.

By applying the principle of dimensional homogeneity, the exponents are determined as

x = 0.5

(for pressure) and

y = -0.5

(for density). Thus,

x - y = 0.5 - (-0.5) = 1.0

Step-by-Step Solution

Determine the dimensions of speed, pressure, and density.

[v] = [L T^{-1}]

[P] = [M L^{-1} T^{-2}]

, and

[\rho] = [M L^{-3}]

Dimensional homogeneity requires expressed physical quantities to be broken down into fundamental dimensions (

M

L

T

Substitute dimensions into the relationship

v = C P^x \rho^y

[L T^{-1}] = [M L^{-1} T^{-2}]^x \, [M L^{-3}]^y = M^{x+y} \, L^{-x-3y} \, T^{-2x}

This establishes a system of algebraic equations by equating exponents of corresponding fundamental dimensions.

Solve for exponents

x

and

y

From time

T

-2x = -1 \implies x = 0.5

. From mass

M

x + y = 0 \implies y = -0.5

Equating the powers of fundamental dimensions on both sides yields the values of

x

and

y

Calculate the required expression

(x - y)

x - y = 0.5 - (-0.5) = 1.0

Subtracting negative

0.5

from

0.5

results in

1.0

Key Concept

Dimensional Analysis and Homogeneity

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