Question

A small steel ball of radius $r$ is allowed to fall under gravity through a column of viscous liquid of coefficient $\eta$ . After some time the velocity of the ball attains a constant value known as terminal velocity, $V_T$. The terminal velocity depends on mass of the ball '$m$', coefficient of viscosity '$\eta$', the radius of the ball '$r$' and acceleration due to gravity $g$. The relationship between terminal velocity and other factors given is :

• A. $V_T\propto \frac{mg}{\eta r}$
• B. $V_T\propto \frac{\eta r}{mg}$
• C. $V_T\propto$ $\eta rmg$
• D. $V_T\propto \frac{mgr}{\eta }$

Answer 1

Answer: (A)

$V_T\propto \frac{mg}{\eta r}$

$V_T\propto m^{\alpha }{\eta }^{\beta }{r}^{\gamma }{g}^k$
$M^0{L}^1{T}^{-1}=\left(M^1{)}^{\alpha }\right(M^1{L}^{-1}{T}^{-1}{)}^{\beta }\left(L^1{)}^{\gamma }\right(L{T}^{-2}{)}^k$
$0=\alpha +\beta \Rightarrow \alpha =1=k$
$1=-\beta +\gamma +k\beta =\gamma =-1$
$-1=-\beta -2k$
$\Rightarrow V_T\propto \frac{mg}{\eta r}$