Question

A spherical solid ball of volume V is made of a material of density ${\rho }_1$ It is falling through a liquid of density ${\rho }_1({\rho }_2<{\rho }_1)$. Assume that the liquid applies a viscous force on the ball that is Proportional to the square of its speed v, i.e., $F_{viscous}=-{kv}^2(k>0)$. The terminal speed of the ball is :

• A. $\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$
• B. $\frac{Vg{\rho }_1}{k}$
• C. $\sqrt{\frac{Vg{\rho }_1}{k}}$
• D. $\frac{Vg({\rho }_2<{\rho }_1)}{k}$

Answer 1

The correct option is A $\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$

The condition for terminal speed $\left(v_t\right)$ is Weight = Buoyant force + Viscous force

$W=V{\rho }_{1}g$

$V{\rho }_{1}g=V{\rho }_{2}g+k{v}_t^2$

$v_t=\sqrt{\frac{Vg({\rho }_1-{\rho }_2)}{k}}$