A spherical ball of mass 4m, density `sigma` and radius `r` is attached to a pulley-mass system as shown in figure. The ball is released in a liquid of coefficient of viscosity `eta` and density `rho(lt(sigma)/(2))`. If the length of the liquid column is sufficiently long, the terminal velocity attained by the ball is given by (assume all pulleys to be massless and string as massless and inextensible):

Options

A. `(2)/(9)(r^2(2sigma-rho)g)/(eta)`
B. `(2)/(9)(r^2(sigma-2rho)g)/(eta)`
C. `(2)/(9)(r^2(sigma-4rho)g)/(eta)`
D. `(2)/(9)(r^2(sigma-3rho)g)/(eta)`

Answer

B

Explanation

From the free body diagram of the sphere:
`F_V=4mg-2mg-F_B`
`impliesF_V=2mg-F_B`
`implies6pietarV=(4)/(3)pir^3((sigma)/(2)-rho)g`
(since `4m=(4)/(3)pir^3xxsigma)`
`impliesV=(2)/(9)r^2((sigma-2rho)g)/(eta)`