Q. 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 $\upsilon_{T}$. The terminal velocity depends on (i) the mass of the ball $m$, (ii) $\eta$ , (iii) $r$ and (iv) acceleration due to gravity $g$. Which of the following relations is dimensionally correct
A) $\upsilon_{T} \propto \frac{mg}{\eta r}$
B) $\upsilon_{T} \propto \frac{\eta r}{mg}$
C) $\upsilon_{T} \propto \eta rmg$
D) $\upsilon_{T} \propto \frac{mgr}{\eta}$
Explanation
By submitting dimension of each quantity in R.H.S. of option (a) we get $\left\lbrack \begin{array}{r} \frac{mg}{\eta r} \\ \ \end{array} \right\rbrack = \left\lbrack \frac{M \times LT^{- 2}\ }{ML^{- 1}T^{- 1} \times L} \right\rbrack = \lbrack LT^{- 1}\rbrack$ This option gives the dimension of velocity
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.