question_answer

Speed of a ball of 2 cm radius in a viscous liquid is 20 cm/s. Then the speed of 1 cm radius of ball in the same liquid is :

A)  80 cm/s                              

B)  40 m/s

C)  10 cm/s                              

D)  5 cm/s

Correct Answer: D

Solution :

                 Let the radius of ball is r and density is \[\rho ,\] falling freely in a liquid whose density is \[\sigma \]  and coefficient of viscosity t|. When it attains a terminal velocity v it is subjected to two forces. (i) Effective force acting downwards \[=V(\rho -\sigma )g=\frac{4}{3}\pi {{r}^{3}}(\rho -\sigma )g\] (ii) Viscous force acting upward = 6nr}rv Since, ball is moving with a constant speed v, there is no acceleration in it, the net force acting on it, must be zero. That is, \[6\,\pi \eta \,rv=\frac{4}{3}\pi {{r}^{3}}(\rho -\sigma )g\] \[\Rightarrow \]    \[v=\frac{2}{9}\frac{{{r}^{2}}(\rho -\sigma )g}{\eta }\] \[\Rightarrow \]      \[v\propto {{r}^{2}}\] Here \[{{v}_{1}}=20\,cm/s,\,{{r}_{1}}=2\,cm,\,{{v}_{2}}=?,\,\,{{r}_{2}}=\,1\,cm\] \[\therefore \]       \[\frac{{{v}_{1}}}{{{v}_{2}}}=\frac{2}{{{v}_{2}}}=\frac{{{(2)}^{2}}}{{{(1)}^{2}}}\] \[\Rightarrow \]    \[{{v}_{2}}=20/4\,=5\,cm/s\]