Question Detail
Question
If force depends on density, velocity and length as F ∝ k ρˣ vʸ lᶻ, where k is a dimensionless constant. Find x, y, z.
Correct Answer
Option B is the correct answer.
Detailed Explanation
Dimensions: F = MLT⁻². ρ = ML⁻³, v = LT⁻¹, l = L. So, MLT⁻² = (ML⁻³)ˣ (LT⁻¹)ʸ (L)ᶻ = Mˣ L⁻³ˣ⁺ʸ⁺ᶻ T⁻ʸ. Equate exponents: M: 1 = x; T: -2 = -y => y = 2; L: 1 = -3x + y + z = -3(1) + 2 + z = -1 + z => z = 2. So (x, y, z) = (1, 2, 2). Option B is (1, 2, 1) which doesn't match. Option D is (1,1,2). None match (1,2,2). Possibly the exponent on l is 2z? The problem says l²ᶻ, then it would be l^(2z), giving L exponent: 1 = -3x + y + 2z. With x=1, y=2: 1 = -3 + 2 + 2z => 1 = -1 + 2z => 2z = 2 => z=1. So then (x,y,z)=(1,2,1) which is option B.
Hint
Use dimensional analysis. Write down the dimensions of each quantity and equate powers of M, L, T.