If terminal speed of a sphere of gold (density = 19.5 kg/m 3 ) is 0.2 m/s in viscous liquid (density = 1.5 kg/m 3 ), find the terminal speed of a sphere of silver (density = 10.5 kg/m 3 ) of the same

Question

If terminal speed of a sphere of gold (density = 19.5 kg/m 3 ) is 0.2 m/s in viscous liquid (density = 1.5 kg/m 3 ), find the terminal speed of a sphere of silver (density = 10.5 kg/m 3 ) of the same size in the same liquid.

TestHub · Accepted Answer

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

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

Since $r$ , $η$ , and $g$ are constant for both spheres, the terminal velocity is directly proportional to the difference in densities:

$v \propto (ρ - σ)$

For the gold sphere:

$v_{1} \propto (ρ_{gold} - σ)$

$0.2 \propto (19.5 - 1.5)$

$0.2 \propto 18$

For the silver sphere:

$v_{2} \propto (ρ_{silver} - σ)$

$v_{2} \propto (10.5 - 1.5)$

$v_{2} \propto 9$

Taking the ratio of the two equations:

$\frac{v _{2}}{0.2} = \frac{9}{18}$

$\frac{v _{2}}{0.2} = \frac{1}{2}$

$v_{2} = 0.1 m/s$

Answer: $0.1 m/s$

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