Question

The terminal speed of a sphere of gold (density = $19.5{\text{g/cm}}^3)$ is $0.2\text{m/s}$ in a viscous liquid (density = $1.5{\text{g/cm}}^3$). Find the terminal speed of a sphere of silver (density $=10.5{\text{g/cm}}^3)$ of the same size in the same liquid (in $\text{m/s}$)

Answer 1

Given, ${\rho }_g=19.5{\text{g/cm}}^3$ (density of gold)
${\rho }_l=1.5{\text{g/cm}}^3$ (density of liquid)
${\rho }_s=10.5{\text{g/cm}}^3$ (density of silver)
Terminal speed of sphere of sphere of gold, $v_g=0.2\text{m/s}$
Let the terminal speed of sphere of silver $=v_s$

Terminal velocity of any body is given by $\left(v\right)=\frac{2({\rho }_b-{\rho }_l)r^{2}g}{9\mu }$
Both sphere of gold & silver are of same size.
So, Terminal velocity $\propto ({\rho }_b-{\rho }_l)$
where ${\rho }_b=$ density of spheres
${\rho }_l$ = density of liquid
$\therefore \frac{v_g}{v_s}=\frac{({\rho }_g-{\rho }_l)}{({\rho }_s-{\rho }_l)}$
$\Rightarrow \frac{0.2}{v_s}=\frac{(19.5-1.5)}{(10.5-1.5)}$
$\Rightarrow v_s=0.1\text{m/s}$