Physics - Fluid Question with Solution | TestHub

Mass of a solid sphere is given by $M=\frac{4}{3}\pi r^3\rho$.

Since both spheres are made of the same material, their density $\rho$ is constant, which implies $M\propto r^3$.

For the two spheres, $\frac{M_2}{M_1}=\frac{8M}{M}=8$.

Therefore, ${\left(\frac{r_2}{r_1}\right)}^3=8\Rightarrow \frac{r_2}{r_1}=2$.

The terminal velocity of a sphere in a viscous liquid is given by $V=\frac{2}{9}\frac{r^2(\rho -\sigma )g}{\eta }$.

Since $\rho$, $\sigma$, and $\eta$ are constant, $V\propto r^2$.

Taking the ratio of terminal velocities, $\frac{V_2}{V_1}={\left(\frac{r_2}{r_1}\right)}^2=(2{)}^2=4$.

Given $V_1=V$ and $V_2=nV$, we get $n=4$.

Answer: $4$