Question Detail
Question
Given that T = k aˣ ρʸ γᶻ, use dimensional analysis to determine the value of x, y and z in the relation, where T is the period of vibration, a is the radius, ρ is the density and γ is the surface tension.
Correct Answer
Option C is the correct answer.
Detailed Explanation
Dimensions: T = [T], a = [L], ρ = [ML⁻³], γ = [MT⁻²]. So [T] = [L]ˣ [ML⁻³]ʸ [MT⁻²]ᶻ = M^(y+z) L^(x - 3y) T^(-2z). Equating: M: y+z = 0; L: x - 3y = 0; T: -2z = 1 → z = -1/2. Then y = -z = 1/2, and x = 3y = 3/2. So (x,y,z) = (3/2, 1/2, -1/2). Option C is (3/2, 3/2, -3/2) which is different. Possibly I misidentified: ρ is density [ML⁻³], γ surface tension [MT⁻²]. Then y+z=0, x-3y=0, -2z=1 → z=-1/2, y=1/2, x=3/2. That's (3/2, 1/2, -1/2). Not in options. If γ is surface tension but sometimes given as [MT⁻²] correct. Option B is (1/2, 3/2, -3/2). Option C is (3/2, 3/2, -3/2). None match. Possibly ρ is density but they used different symbols.
Hint
Write dimensional equations and solve for exponents.