A spherical solild of volume V is made of a material of density `rho_(1)`. It is falling through a liquid of density `rho_(2)(rho_(2) lt rho_(1))`. Assume that the liquid applies a viscous froce on the ball that is proportional ti the its speed v, i.e., `F_(viscous)=-kv^(2)(kgt0)`. The terminal speed of the ball is

Options

A. `(Vg (rho_(1) - rho_(2)))/(k)`
B. `sqrt((Vg (rho_(1) - rho_(2)))/(k))`
C. `(Vg rho_(1))/(k)`
D. `sqrt((V g rho_(1))/(k))`

Answer

B

Explanation

The forces acting on the solid ball when it is falling through a liquid are mg downwards, upthrust upwards and the viscous force also acting upwards. The viscous force rapidly increases with velocity, attaining a maximum when the ball reaches the terminal velocity. Then the acceleration is zero. `mg- V rho_(2) g - kv^(2) = ma` where V is volume, v is the terminal velocity. When the ball is moving with terminal velocity, a= 0
`:. V rho_(1) g- V rho_(2)g - kv^(2) = 0`
or `v^(2) = (vg (rho_(1) - rho_(2)))/(k) or v= sqrt((vg (rho_(1) - rho_(2)))/(k))`