Question

A small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time, the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on $\left(i\right)$ the mass of the ball $m$, $\left(ii\right)$ $\eta$, $\left(iii\right)r$ and $\left(iv\right)$ acceleration due to gravity $g$. . Which of the following relations is dimensionally correct?

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

Answer 1

The correct option is C $v_T\propto \frac{mg}{\eta r}$

First, we check option (a)
Substitute dimensions of each quantity in $R.H.S.$ of option (a)
$\left[\frac{mg}{\eta r}\right]=\frac{\left[M^1\right]\left[L^1{T}^{-2}\right]}{\left[M^1{L}^{-1}{T}^{-1}\right]\left[L^1\right]}$

$=\left[M^0{L}^1{T}^{-1}\right]$

This is dimension of velocity, so option is dimensionally correct.
Final answer: (a)