A-small-solid-spherical-ball-of-high-density-is-dropped-in-a-viscous-liquid-Its-journey-in-the-liquid-is-best-described-in-the-following-figure-by-the-curve-

Question Number 20973 by Tinkutara last updated on 09/Sep/17

A small solid spherical ball of high

density is dropped in a viscous liquid .

Its journey in the liquid is best described

in the following figure by the curve

Commented by Tinkutara last updated on 09/Sep/17

Answered by dioph last updated on 09/Sep/17

F = m \frac{d^{2} x}{{d t}^{2}}

Let b \equiv 6 π η R

m g - b \frac{d x}{d t} = m \frac{d^{2} x}{{d t}^{2}}

\frac{d^{2} x}{{d t}^{2}} + \frac{b}{m} \frac{d x}{d t} = g

x (t) = K_{1} t + K_{2} + K_{3} e^{- b t / m}

K_{1} = \frac{m g}{b}

\frac{d x}{d t} ∣_{0} = 0 \Leftrightarrow \frac{m g}{b} - \frac{b}{m} K_{3} = 0

\Rightarrow K_{3} = \frac{m^{2} g}{b^{2}}

x (0) = 0 \Leftrightarrow K_{2} + K_{3} = 0

\Rightarrow K_{2} = \frac{- m^{2} g}{b^{2}}

x (t) = \frac{m g}{b} (t - \frac{m}{b} (1 - e^{- b t / m}))

\frac{d x}{d t} = \frac{m g}{b} (1 - e^{- b t / m})

In steady - state, v = \frac{m g}{b} (constant)

So answer should be curve C

Commented by Tinkutara last updated on 09/Sep/17

Thank you very much Sir!