Question

A sphere is dropped under gravity through a fluid of viscosity `eta`. Taking the average ac celeration as half of the initial ac celeration, show that the time to attain the terminal velocity is independent of the fluid density.

Answer 1

Let r be the radius of the sphere and `rho` be its density.
It is dropped in a flui of density `rho_(0)` . On just entering into the fluid, the effective downward froce acting on the sphere is
`F = mass xx ac cel erati on = 4/3 pi r^(3) (rho -rho_(0))g`
Let a be the initial ac celeration produced. then,
`a=(f o r ce)/(mass) = (4/3 pi r^(3)(rho -rho_(0))g)/(4/3 pi r^(3) rho)= ((rho-rho_(0))/(rho))g`
When the sphere attains the terminal velocity `upsilon` its ac celeration becomes zero.
`:.` average ac celeration,
`a_(1) =(a+0)/(2) = a/2 = ((rho-rho_(0))/(2rho))g`
Let sphere takes time t to attain the terminal velocity `upsilon` when dropped in the fluid. then `u=0`, using the relation , `upsilon= u +a_(1)t`, we get
`(2r^(2) (rho-rho_(0))g)/(9 eta) = 0 + ((rho-rho_(0))/(2rho))g t`
or ` t= (2r^(2) (rho-rho_(0))g)/(9 eta)* (2rho)/((rho-rh_(0))g) = (4rho r^(2))/(9 eta)`
From this relation, it is clear that time t is independent of the density `rho_(0)` of the given fluid.