A spherical solid ball of volume V is made of a material of density ρ₁ . It is falling through a

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9-1.Fluid Mechanics

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A spherical solid ball of volume $V$ is made of a material of density $\rho_1$. It is falling through a liquid of density $\rho_1 (\rho_2 < \rho_1)$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{viscous} = -kv^2 (k > 0)$. The terminal speed of the ball is

A

$\sqrt {\frac{{Vg\left( {{\rho _1} - {\rho _2}} \right)}}{k}}$

B

$\frac{{Vg{\rho _1}}}{k}$

C

$\sqrt {\frac{{Vg{\rho _1}}}{k}}$

D

$\frac{{Vg\left( {{\rho _1} - {\rho _2}} \right)}}{k}$

Solution

The condition for terminal speed $\left( {{v_t}} \right)$ is Weight $=$ Buoyant force $+$ Viscous force

$\therefore \mathrm{V} \rho_{1} \mathrm{g}=\mathrm{V} \rho_{2} \mathrm{g}+\mathrm{kv}_{t}^{2}$

$\therefore \mathrm{v}_{\mathrm{t}}=\sqrt{\frac{\mathrm{Vg}\left(\rho_{1}-\rho_{2}\right)}{\mathrm{k}}}$

Standard 11

Physics

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