Question Detail
Question
If the force F = k ρᵃ vᵇ Aᶜ, where ρ is the density of air, v is the velocity of motion, A is the cross sectional area and k is a dimensionless constant. Find a, b, c.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Force F has dimensions [MLT⁻²]. ρ has [ML⁻³], v has [LT⁻¹], A has [L²]. So [MLT⁻²] = [ML⁻³]ᵃ [LT⁻¹]ᵇ [L²]ᶜ = Mᵃ L⁻³ᵃ⁺ᵇ⁺²ᶜ T⁻ᵇ. Equate: M: a = 1; T: -b = -2 → b = 2; L: -3a + b + 2c = 1 → -3 + 2 + 2c = 1 → -1 + 2c = 1 → 2c = 2 → c = 1. So (a,b,c) = (1,2,1). Not in options. Option A is 1,0.5,-0.5. That would give L: -3 + 0.5 -1 = -3.5, not 1. So none match. Possibly the formula is for drag force F = ½ρv²A, so a=1, b=2, c=1.
Hint
Use dimensional analysis.