Fluid Mechanics

JEE Main level, single correct, medium

190 of 192

Question 190 of 192

A spherical solid ball of volume VV is made of a material of density ρ1\rho_1. It is falling through a liquid of density ρ2(ρ2<ρ1)\rho_2\left(\rho_2 \lt \rho_1\right). Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed v, i.e., Fviscous =−kv2(k>0)F_{\text {viscous }}=-k v^2(k\gt 0). The terminal speed of the ball is

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Answer: (A) Vg⁡(ρ1−ρ2)k\sqrt{\frac{\operatorname{Vg}\left(\rho_1-\rho_2\right)}{\mathrm{k}}}

ρ1Vg−ρ2Vg=kvT2⇒VT=Vg(ρ1−ρ2)k\begin{aligned} & \rho_1 V g-\rho_2 V g=k v_T^2 \\ & \Rightarrow \quad V_T=\sqrt{\frac{V g\left(\rho_1-\rho_2\right)}{k}} \end{aligned}

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