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Q.

The critical angular velocity w of a cylinder inside another cylinder containing a liquid at which its turbulence occurs depends on viscosity $η$ , density d and the distance x between the walls of the cylinders. Then w is proportional to

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a

$\frac{η^{2}}{xd}$

b

$\frac{η}{d^{2} x}$

c

$\frac{η}{x^{2} d}$

d

$\frac{xd}{η}$

answer is A.

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Detailed Solution

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$\begin{array}{l} {ωαη}^{a} d^{b} X^{c} \\ T^{- 1} \propto {({ML}^{- 1} T^{- 1})}^{a} {({ML}^{- 3})}^{b} L^{c} \\ T^{- 1} \propto M^{a + b} L^{- a - 3 b + c} T^{- a} \\ - a = - 1 \Rightarrow a = 1 \\ a + b = 0 \Rightarrow b = - 1 \\ - a - 3 b + c = 0 \Rightarrow c = - 2 \\ ω \propto \frac{η}{{dx}^{2}} \end{array}$

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