Question

A spherical body of mass m and radius ris allowed to fall in a medium of viscosity $\eta$. The time in which the velocity of the body increases from zero to 0.63 times the terminal velocity (v) is called time constant (t). Dimensionally $\tau$ can be represented by:

• A. $\frac{m{r}^2}{6\pi \eta }$
• B. $\sqrt{\frac{6\pi mm}{g^2}}$
• C. $\sqrt{\frac{m}{6\pi \eta rv}}$
• D. None of these

Answer 1

Answer: (D)

None of these

Option A has dimension : $\frac{M{L}^2}{M{L}^{-1}{T}^{-2}}=L^3{T}^2$

Option $B$ has dimension : $\sqrt{\frac{\left(ML\right)\left(M{L}_{-1}{T}^{-2}\right)}{L^2{T}^{-4}}}=M{L}^{-1}T$

Option C has dimension : $\frac{M}{\left(M{L}^{-1}{T}^{-2}\right)\left(L\right)(L{T}^{-1}}=\frac{T^3}{L}$

For options (1), (2) and (3) dimensions are not matching with time constant. Hence the answer will be none of the above.