Dimensional Analysis ApplicationsMediumQuestion 5024
Question
Assume a sound speed depends only on pressure P and density ρ through v = kPᵃρᵇ, with dimensionless k. Which dependence is dimensionally consistent?
- A√(P/ρ)Correct
- B√(Pρ)
- CP/ρ
- DP²/ρ
Correct answer
√(P/ρ)
Explanation
[v] = LT⁻¹, [P] = ML⁻¹T⁻², [ρ] = ML⁻³. If v = kPᵃρᵇ: LT⁻¹ = (ML⁻¹T⁻²)ᵃ(ML⁻³)ᵇ gives a = 1/2, b = −1/2, so v ∝ √(P/ρ).