Question

What is the dimensional formula of coefficient of viscosity?
\[
  {\text{A}}{\text{. }}\left[ {{\text{ML}}{{\text{T}}^{ - 1}}} \right] \\
  {\text{B}}{\text{. }}\left[ {{{\text{M}}^{ - 1}}{{\text{L}}^2}{{\text{T}}^{ - 2}}} \right] \\
  {\text{C}}{\text{. }}\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 1}}} \right] \\
 \]
\[{\text{D}}{\text{.}}\] None

Answer 1

Hint: Here, we will proceed by expressing the coefficient of viscosity in terms quantities whose dimensional formulas are known which can be done by using Newton’s law of viscosity.

Step By Step Answer:

Formula Used- ${\text{F}} = \eta \dfrac{{d{\text{u}}}}{{dy}}$.

According to Newton’s law of viscosity

Force of friction ${\text{F}} = \eta \dfrac{{d{\text{u}}}}{{dy}}$ where $\eta $ denotes the coefficient of viscosity, du denotes the change in the velocity of the fluid layers separated by a distance dy (measured in the vertical direction)

\[
   \Rightarrow \eta = \dfrac{{\text{F}}}{{\left( {\dfrac{{d{\text{u}}}}{{dy}}} \right)}} \\
   \Rightarrow \eta = {\text{F}}\left( {\dfrac{{dy}}{{d{\text{u}}}}} \right){\text{ }} \to {\text{(1)}} \\
 \]

As we know that F represents a force and the dimension of force is \[\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\]. Also, dy represents the length and the dimension of length is [L]. Also, du represents the change in the velocity and the dimension of velocity is \[\left[ {{\text{L}}{{\text{T}}^{ - 1}}} \right]\].

Using the formula given by equation (1), the dimension of the coefficient of viscosity is given by

Dimensional formula of coefficient of viscosity =

(Dimensional formula of F)($\dfrac{{{\text{Dimensional formula of dy}}}}{{{\text{Dimensional formula of du}}}}$)

$ \Rightarrow $ Dimensional formula of coefficient of viscosity =
(Dimensional formula of force)($\dfrac{{{\text{Dimensional formula of length}}}}{{{\text{Dimensional formula of velocity}}}}$)

$ \Rightarrow $ Dimensional formula of coefficient of viscosity = \[\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\dfrac{{\left[ {\text{L}} \right]}}{{\left[ {{\text{L}}{{\text{T}}^{ - 1}}} \right]}} = \left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\left[ {\text{L}} \right]\left[ {{{\text{L}}^{ - 1}}{{\text{T}}^1}} \right]\]

$ \Rightarrow $ Dimensional formula of coefficient of viscosity = \[\left[ {{\text{ML}}{{\text{T}}^{ - 2}}{\text{L}}{{\text{L}}^{ - 1}}{\text{T}}} \right] = \left[ {{\text{M}}{{\text{L}}^{1 + 1 - 1}}{{\text{T}}^{ - 2 + 1}}} \right] = \left[ {{\text{M}}{{\text{L}}^1}{{\text{T}}^{ - 1}}} \right]\]

$ \Rightarrow $ Dimensional formula of coefficient of viscosity = \[\left[ {{\text{ML}}{{\text{T}}^{ - 1}}} \right]\]

Therefore, option A is correct.

Note: Coefficient of viscosity refers to the measurement of the fluid's viscosity, equal to the force per unit area required to maintain a velocity difference of one-unit distance per unit time between two parallel fluid planes in the flow direction divided by one-unit distance. They are usually expressed in poise or centipoise.