Question

What is the dimensional formula of coefficient of viscosity?
$$\begin{gathered} \text{A}\text{. }\left[\text{ML}{\text{T}}^{-1}\right] \\ \text{B}\text{. }\left[{\text{M}}^{-1}{\text{L}}^2{\text{T}}^{-2}\right] \\ \text{C}\text{. }\left[\text{M}{\text{L}}^{-1}{\text{T}}^{-1}\right] \end{gathered}$$
$$\text{D}\text{.}$$ 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- $\text{F}=\eta {\displaystyle \frac{d\text{u}}{dy}}$.

According to Newton’s law of viscosity

Force of friction $\text{F}=\eta {\displaystyle \frac{d\text{u}}{dy}}$ 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)

$$\begin{gathered} \Rightarrow \eta ={\displaystyle \frac{\text{F}}{\left({\displaystyle \frac{d\text{u}}{dy}}\right)}} \\ \Rightarrow \eta =\text{F}\left({\displaystyle \frac{dy}{d\text{u}}}\right)\to \text{(1)} \end{gathered}$$

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)(${\displaystyle \frac{\text{Dimensional formula of dy}}{\text{Dimensional formula of du}}}$)

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

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

$\Rightarrow$ 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.