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 $v_r$. The terminal velocity depends on $\left(i\right)$ the mass of the ball $m,\left(ii\right)\eta ,\left(iii\right)rand\left(iv\right)$ acceleration due to gravity $g$ . Which of the following relations is dimensionally correct

• 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

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

By substituting dimension of each quantity in R.H.S. of option (a) we get $\left[\frac{mg}{\eta r}\right]=\left[\frac{M\times L{T}^{-2}}{M{L}^{-1}{T}^{-1}\times L}\right]$
$=\left[L{T}^{-1}\right].$
This option gives the dimension of velocity.