Question

A sphere of radius $R$ and density ${\rho }_1$ is dropped in a liquid of density $\sigma$. Its terminal velocity is $v_1$ . If another sphere of radius $R$ and density ${\rho }_2$ is dropped in the same liquid, its terminal velocity will be:

• A. $\left(\frac{{\rho }_2-\sigma }{{\rho }_1-\sigma }\right)v_1$
• B. $\left(\frac{{\rho }_1-\sigma }{{\rho }_2-\sigma }\right)v_1$
• C. $\left(\frac{{\rho }_1}{{\rho }_2}\right)v_1$
• D. $\left(\frac{{\rho }_2}{{\rho }_1}\right)v_1$

Answer 1

The correct option is A $\left(\frac{{\rho }_2-\sigma }{{\rho }_1-\sigma }\right)v_1$

Terminal velocity of a sphere of density ${\rho }_1$ falling through a liquid of density $\sigma$ is,
$(V_T{)}_1=\frac{2}{9}\frac{R^{2}g({\rho }_1-\sigma )}{\eta }$
Similarly, terminal velocity of a sphere (radius $R$) of density ${\rho }_2$ falling through liquid of density $\sigma$ is,
$(V_T{)}_2=\frac{2}{9}\frac{R^{2}g({\rho }_2-\sigma )}{\eta }$
On taking the ratio,
$\frac{(V_T{)}_1}{(V_T{)}_2}=\frac{({\rho }_1-\sigma )}{({\rho }_2-\sigma )}$
$\Rightarrow (V_T{)}_2=\frac{({\rho }_2-\sigma )}{({\rho }_1-\sigma )}(V_T{)}_1$
Since $(V_T{)}_1=v_1$,
$(V_T{)}_2=\frac{({\rho }_2-\sigma )}{({\rho }_1-\sigma )}{v}_1$