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A spherical body of mass and radius is allowed to fall in a medium of viscosity. The time in which the velocity of the body increases from zero to $0.63$ times the terminal velocity is called time constant. Dimensionally can be represented by
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None of the above
Time constant $\tau=[T]$ and Viscosity $\eta=\left[M {L}^{-1} T^{-1}\right]$
For options (a), (b) and (c) dimensions are not matching with time constant.
Option A has dimensions: \(\frac{\mathrm{ML}^2}{\mathrm{ML}^{-1} \mathrm{~T}^{-2}}=\mathrm{L}^3 \mathrm{~T}^2\)
Option B has dimensions: \(\sqrt{\frac{(\mathrm{ML}) *\left(\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right)}{\mathrm{L}^2 \mathrm{~T}^{-4}}}=\mathrm{ML}^{-1} \mathrm{~T}\)
Option C has dimensions: \(\frac{\mathrm{M}}{\left(\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right) *(\mathrm{~L}) *\left(\mathrm{LT}^{-1}\right)}=\frac{\mathrm{T}^3}{\mathrm{~L}}\)
For options (a), (b) and (c) dimensions are not matching with time constant.
Option A has dimensions: \(\frac{\mathrm{ML}^2}{\mathrm{ML}^{-1} \mathrm{~T}^{-2}}=\mathrm{L}^3 \mathrm{~T}^2\)
Option B has dimensions: \(\sqrt{\frac{(\mathrm{ML}) *\left(\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right)}{\mathrm{L}^2 \mathrm{~T}^{-4}}}=\mathrm{ML}^{-1} \mathrm{~T}\)
Option C has dimensions: \(\frac{\mathrm{M}}{\left(\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right) *(\mathrm{~L}) *\left(\mathrm{LT}^{-1}\right)}=\frac{\mathrm{T}^3}{\mathrm{~L}}\)
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