A sphere of radius r and density ρ is dropped under gravity through a fluid of viscosity η . The average acceleration (till is attains terminal velocity) is half of initial acceleration. If time requi

Question

A sphere of radius r and density ρ is dropped under gravity through a fluid of viscosity η . The average acceleration (till is attains terminal velocity) is half of initial acceleration. If time required to attain terminal velocity is N η 4 r 2 ρ ​ , The value of N is :-

TestHub · Accepted Answer

Initially V = 0, so the viscous force is zero: $a_{0} = \frac{( ρ - ρ _{liq} )}{ρ} g, a_{avg} = \frac{a _{0}}{2}$

At terminal velocity: $m g = F_{B} + 6 π ηr v_{T} \Rightarrow v_{T} = \frac{2 r ^{2} g ( ρ - ρ _{liq} )}{9 η}$

$a_{avg} = \frac{v _{T}}{t} \Rightarrow t = \frac{2 v _{T}}{a _{0}} = \frac{4 r ^{2} ρ}{9 η} \Rightarrow N = 9$

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