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Turpentine oil is flowing through a tube of length ℓ and radius r. The pressure difference between the two ends of the tube is p. The viscosity of oil is given by \( η = \frac{p(r^2 - x^2)}{4ℓv} \) where v is the velocity of oil at a distance x from the axis of the tube. The dimensions of η are

1. [M⁰L⁰T⁰]
2. [ML⁻¹]
3. [ML⁻²T⁻²]
4. [ML⁻¹T⁻¹]

Correct answer: [ML⁻¹T⁻¹]

Solution

Viscosity (η) is defined as the ratio of shear stress to the velocity gradient. Using dimensional analysis, pressure (p) has dimensions [ML⁻¹T⁻²], length (ℓ) has [L], radius (r) has [L], and velocity (v) has [LT⁻¹]. Substituting these into the given formula, η has dimensions [ML⁻¹T⁻¹].

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