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The terminal velocity vₜ of a small steel ball of radius r falling under gravity through a column of a viscous liquid of coefficient of viscosity η depends on mass of the ball m, acceleration due to gravity g, coefficient of viscosity η and radius r. Which of the following relations is dimensionally correct?

1. vₜ ∝ η / r
2. vₜ ∝ r² / η
3. vₜ ∝ η / r²
4. vₜ ∝ r / η

Correct answer: vₜ ∝ r² / η

Solution

Using dimensional analysis, terminal velocity vₜ has dimensions of [L T⁻¹]. The relation vₜ ∝ r² / η is dimensionally consistent because the dimensions of r² / η match [L T⁻¹].

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