Question

The dimension of coefficient of viscosity.
A. $\left[ MLT \right]$
B. \[\left[ M{{L}^{-1}}{{T}^{-1}} \right]\]
C. \[\left[ ML{{T}^{-1}} \right]\]
D. \[\left[ ML{{T}^{-2}} \right]\]

Answer 1

Hint: To write dimensional formula for any quantity we need formula for that. Then we write formulas in basic terms like mass , time, length etc.
We can define coefficient of viscosity with following formula as below:
\[\eta =\dfrac{\tan gential\,force\,\times \,dis\tan ce\,\,between\,layers}{area\,of\,layer\times \,velocity}\]
Where \[\eta \] is coefficient of viscosity.

Complete step-by-step answer:
We generally define coefficient of viscosity as resistance which a fluid can exerts against a flow caused by applied force.
Formula of coefficient of viscosity is
\[\eta =\dfrac{\tan gential\,force\,\times \,dis\tan ce\,\,between\,layers}{area\,of\,layer\times \,velocity}\]
Now we can write dimensional formulas for each quantity.
Dimensional formula for force is $\left[ ML{{T}^{-2}} \right]$ because $F=ma$
Dimensional formula for distance is $\left[ L \right]$
Dimensional formula for the area of the layer is $\left[ {{L}^{2}} \right]$ because the area is generally defined as a centimeter square.
Dimensional formula for velocity is $\left[ L{{T}^{-1}} \right]$ because velocity is defined as ratio of distance and time.
Now we can find dimensional formula for coefficient of viscosity is
\[\Rightarrow \eta =\dfrac{\left[ ML{{T}^{-2}} \right]\,\times \,\left[ L \right]}{\left[ {{L}^{2}} \right]\times \,\left[ L{{T}^{-1}} \right]}\]
\[\Rightarrow \eta =\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]\,}{\,\left[ {{L}^{3}}{{T}^{-1}} \right]}\]
\[\Rightarrow \eta =\left[ M{{L}^{2-3}}{{T}^{-2+1}} \right]\]
\[\Rightarrow \eta =\left[ M{{L}^{-1}}{{T}^{-1}} \right]\]
Hence option B is correct.

Note: To simplify dimensional formulas we can apply multiplication and division properties of exponent.
According to the multiplication property of exponent if we have exponent terms of same base in multiplication then we can add their exponent. We can write it as below:
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
According to the division property of exponent if we have exponent terms of the same base in division then we can subtract their exponent. We can write it as below:
${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$