If dimensions of critical velocity υc of a liquid flowing

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If dimensions of critical velocity υ_c of a liquid flowing through a tube are expressed as , [ η^x ρ^y r^x ] where η, ρ and r are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of x, y and z are given by :

1. – 1, – 1, 1
2. – 1, – 1, – 1
3. 1, 1, 1
4. 1, – 1, – 1

Correct Option: D

Applying dimensional method :
v_c = η^xρ^y r^z
[M⁰LT^{– 1}] = [ML^{– 1}T^{– 1}]^x [ML^{– 3}T⁰]^y [M⁰LT⁰]^z
Equating powers both sides
x + y = 0; – x = – 1 ∴ x = 1
1 + y = 0 ∴ y = – 1
– x – 3y + z = 1
– 1 – 3( – 1) + z = 1
– 1 + 3 + z = 1
∴ z = – 1