One of the available answers from below has no units because it is a "dimensionless ratio" (a ratio that does not have units-i.e. the units of the numerator and denominator should cancel each other out...). Which of the following represents a dimensionless ratio relevant to a moving circular blob of fluid where the fluid density is ρ, the blob diameter is D and at a given instant in time the blob parcel has a velocity U and an acceleration, A.

Work in SI base units.

Question

One of the available answers from below has no units because it is a "dimensionless ratio" (a ratio that does not have units-i.e. the units of the numerator and denominator should cancel each other out...). Which of the following represents a dimensionless ratio relevant to a moving circular blob of fluid where the fluid density is ρ, the blob diameter is D and at a given instant in time the blob parcel has a velocity U and an acceleration, A.

Work in SI base units.

BlackTom AI · Accepted Answer

Consider the problem's goal: form a dimensionless ratio using the given quantities related to a moving circular blob, with dimensions in SI base units.
Option 1: [D^3U^2]/(A D^4). Break down the dimensions: D has dimension L, U has dimension L/T, A has dimension L/T^2. So the numerator D^3 U^2 gives L^3 * (L^2/T^2) = L^5 / T^2. The denominator A D^4 gives (L/T^2) * L^4 = L^5 / T^2. The ratio of these two quantities has dimensions (L^5/T^2)/(L^5/T^2) = 1, i.e., it is dimensionless. This option cleanly cancels all length, time terms, leaving no remaining fundamental units.
Option 2: None of the available options. This would only be correct if every listed expression failed to be dimensionless or had inconsistent units; however, we just found a dimensionless combination among the options, so this choice is not accurate as a general statement about all options.
Option 3: [D^3U^2]/(ρ A D^4). Here, take the same numerator as in option 1, L^5/T^2. The denominator adds ρ A D^4, whose dimensions are ρ (M/L^3) times A (L/T^2) times D^4 (L^4) → M * L^2 / T^2. Therefore the overall dimensions become (L^5/T^2) / (M L^2 / T^2) = L^3 / M. This leaves mass as a residual unit, so it is not dimensionless. Such a ratio would not qualify as a pure dimensionless quantity.
Option 4: [ρ D^2 U^2]/(ρ A D^4). The ρ cancels between numerator and denominator, leaving (D^2 U^2)/(A D^4). The numerator D^2 U^2 has dimensions L^2 * (L^2/T^2) = L^4 / T^2. The denominator A D^4 has dimensions (L/T^2) * L^4 = L^5 / T^2. The ratio is (L^4/T^2)/(L^5/T^2) = 1/L, i.e., it has a length dimension, not dimensionless. Thus this option fails to be dimensionless.
Option 5: [ρ D^2 U^2]/(ρ D^4). Here ρ cancels as well, giving (D^2 U^2)/(D^4) = U^2/D^2. U^2 has dimensions L^2/T^2, D^2 is L^2, so the result is (L^2/T^2)/(L^2) = 1/T^2, which is a time inverse squared unit, not dimensionless. Therefore not suitable as a dimensionless ratio.
In summary, the dimensional analysis shows that only the expression in Option 1 yields a true dimensionless quantity when all given symbols are substituted with their SI base-unit dimensions. The remaining options introduce mass, length, or time dependencies that prevent full cancellation to a dimensionless form.

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