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

Explanation

Let a sphere of radius r and density p falla in a fluid of density p. and viscosity `eta.`
On entering the fluid, the net downward force on the sphere is
F=weight of the sphere - weight of displaced fluid.
`=4/3 pi r^(3) pg-4/3 pi r^(3) pg `
`=4/3 pi r^(3) (p-p.) g`
Initial accerleration of sphere is `a=F/m=(4/3 pi r^(3) (p-p.) g)/(4/3 pi r^(3)p)=((p-p.)/(p))g`
On attaining the terminal velocity, the acceleration of sphere becomes zero.
Average acceleration of sphere `=(a+0)/(2)`
`=((p-p.)/(2p))g`
If the time taken by the sphere to attain the terminal velocity `v=2/9 r^(2)/n (p-p.)g` is t, then
v=u+at
Here, u=0
`rArr 2/9 r^(2)/eta (p-p.) g=((p-p.)/(2p)) gt`
`rArr t=4/9 (r^(2)p)/(eta)`