A solid ball of density `rho_(1)` and radius r falls vertically through a liquid of density `rho_(2)`. Assume that the viscous force acting on the ball is `F = krv`, where `k` is a constant and v its velocity. What is the terminal velocity of the ball ?

Options

A. ` (4pi r ^2 g ( rho _1 - rho _2 ))/(3 k ) `
B. ` (2pi r ( rho_1 - rho_2 ))/(3 g k ) `
C. ` (2pi g ( rho_1 + rho_2 ))/(3 gr ^2 k ) `
D. ` (2pi r ^2 g (rho_1 - rho_2))/(3k ) `

Answer

A

Explanation

Net force on the ball = 0 (when terminal velocity is attained)
Hence, Weight = upthrust + viscous force ` (4)/(3) pi r ^ 3 rho _ 1 g = (4)/(3) pi r ^ 3 rho _2 g + kr v _T , v_T = (4pi g r ^ 2 )/( 3 k ) (rho _ 1 - rho _ 2 ) `