A spherical solid ball of volume V is made of a material of density {ρ ₁} . It is falling t

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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 ${\rho _2}\left( {{\rho _2} < {\rho _1}} \right)$. Assume that the liquid applies a viscous force on the ball that is propoertional to the square of its speed $v$ , i.e., ${F_{{\rm{viscous}}}} = - k{v^2}\left( {k > 0} \right)$. Then terminal speed of the bal 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

$\therefore V \rho_{1} g=V \rho_{2} g+k v_{t}^{2} \therefore v_{t}=\sqrt{\frac{V g\left(\rho_{1}-\rho_{2}\right)}{k}}$

Standard 11

Physics

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