A metal sphere of radius R , density ρ 1 moves with terminal velocity V 1 through a liquid of density σ . Another sphere of same radius but density ρ 2 - moves through same liquid. Its terminal

Question

A metal sphere of radius R , density ρ 1 ​ moves with terminal velocity V 1 ​ through a liquid of density σ . Another sphere of same radius but density ρ 2 ​ - moves through same liquid. Its terminal velocity is V 2 ​ . The ratio V 1 ​ : V 2 ​ is

TestHub · Accepted Answer

The terminal velocity of a sphere of radius $R$ and density $ρ$ falling through a liquid of density $σ$ and coefficient of viscosity $η$ is given by

$V = \frac{2}{9} \frac{R ^{2} ( ρ - σ ) g}{η}$

For the first sphere, the terminal velocity is

$V_{1} = \frac{2}{9} \frac{R ^{2} ( ρ _{1} - σ ) g}{η}$

For the second sphere, the terminal velocity is

$V_{2} = \frac{2}{9} \frac{R ^{2} ( ρ _{2} - σ ) g}{η}$

Taking the ratio of the two terminal velocities, we get

$\frac{V _{1}}{V _{2}} = \frac{ρ _{1} - σ}{ρ _{2} - σ}$

Thus, $V_{1} : V_{2} = (ρ_{1} - σ) : (ρ_{2} - σ)$

Answer: $(ρ_{1} - σ) : (ρ_{2} - σ)$

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