Question

A spherical body of mass $m$ and radius $r$ is allowed to fall in $\alpha$ medium of viscosity $\eta$. The time in which the velocity of the body increases from $0$ to $0.63$ times the terminal velocity $v$ called time constant $\tau$. Dimensionally $\tau$ can be represented by

• A. $\frac{m}{6\pi \eta rv}$
• B. $\sqrt{\left(\frac{6\pi mr\eta }{g^2}\right)}$
• C. $\frac{m{r}^2}{6\pi \eta }$
• D. None of the above

Answer 1

The correct option is D None of the above

$\tau$ is time constant will have Dimension $\left[T\right]$
Solved the question by checking the option,
Option A has dimensions,
$\frac{M{L}^2}{M{L}^{-1}{T}^{-1}}=L^3{T}^1$

Option B has dimensions,
$\sqrt{\frac{\left[ML\right]\left[M{L}^{-1}{T}^{-2}\right]}{\left[L^2{T}^{-4}\right]}}=M{L}^{-1}T$

Option C has dimensions,
$\frac{M}{\left[M{L}^{-1}{T}^{-1}\right]\left[L\right]\left[L{T}^{-1}\right]}=\frac{T^3}{L}$

Final answer: (d)