A small steel ball of radius r is allowed to fall under gravity t… - Vidyadip

MCQ

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 $(i)$ 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

✓
${v_T} \propto \frac{{mg}}{{\eta r}}$
B
${v_T} \propto \frac{{\eta r}}{{mg}}$
C
${v_T} \propto \eta rmg$
D
${v_T} \propto \frac{{mgr}}{\eta }$

✓

Answer

Correct option: A.

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

a
$V_{T} \propto m^{\alpha} \eta^{\beta} r^{\gamma} g^{k}$

$M^{0} L^{1} T^{-1}=\left(M^{1}\right)^{\alpha}\left(M^{1} L^{-1} T^{-1}\right)^{\beta}\left(L^{1}\right)^{\gamma}\left(L T^{-2}\right)^{k}$

$0=\alpha+\beta \Rightarrow \alpha=1=k$

$1=-\beta+\gamma+k \beta=\gamma=-1$

$-1=-\beta-2 k$

$\Rightarrow V_{T} \propto \frac{m g}{\eta r}$

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