Question

The dimensional formula for coefficient of kinematic viscosity is :
A. ${{M}^{0}}{{L}^{-1}}{{T}^{-1}}$
B. ${{M}^{0}}{{L}^{2}}{{T}^{-1}}$
C. $M{{L}^{2}}{{T}^{-1}}$
D. $M{{L}^{-1}}{{T}^{-1}}$

Answer 1

Hint: First we will define what kinematic viscosity is and then we will use the expression in which it is used and accordingly, we will use the dimensional formula for all the other quantities and finally we will determine the dimensional formula for kinematic viscosity using all that.

Complete answer:
Dynamic viscosity gives us the force needed to make a fluid flow at a certain rate whereas kinematic viscosity is the opposite of that, it gives us how fast a fluid will flow when provided with some force. Viscosity is defined as the ratio of shear stress to the velocity gradient of the liquid and kinematic viscosity is defined as the ratio of viscosity and the density of the liquid. The dimensional formula of shear stress is the same as that of pressure as both are defined as the force per unit area. And gradient of liquid has the dimensional formula of velocity divided by length. Hence, the dimensional formula for viscosity will become

\[\dfrac{\left( \dfrac{F}{A} \right)}{\left( \dfrac{v}{l} \right)}=\dfrac{\left( \dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ {{L}^{2}} \right]} \right)}{\left( \dfrac{\left[ L{{T}^{-1}} \right]}{\left[ L \right]} \right)}=\dfrac{\left[ M{{L}^{-1}}{{T}^{-2}} \right]}{\left[ {{T}^{-1}} \right]}=\left[ M{{L}^{-1}}{{T}^{-1}} \right]\]

And this when divided by the dimensional formula for density of the liquid will give the dimensional formula for the kinematic viscosity of the fluid.

\[\dfrac{\left[ M{{L}^{-1}}{{T}^{-1}} \right]}{\left[ M{{L}^{-3}} \right]}=\left[ {{L}^{2}}{{T}^{-1}} \right]=\left[ {{M}^{0}}{{L}^{2}}{{T}^{-1}} \right]\]
This will be the dimensional formula for the kinematic viscosity.

So, the correct answer is “Option B”.

Note:
There are differences in kinematic viscosity and dynamic viscosity as have been mentioned and they will have different units and different dimensional formulas unlike for friction where all the friction coefficients have the same units and dimensional formula.