Question

Which of the following is unitless quantity?
A. Pressure gradient
B. Displacement gradient
C. Force gradient
D. Velocity gradient

Answer 1

Hint: Here, in this question, we will first try to understand the concept behind the term gradient and then, determine the expression for each of the four given gradient terms in the options to get to our final result.

Formula Used:
Gradient of a function ‘f’ is written as,
\[\nabla f = \dfrac{{df}}{{dx}}\]
where,
\[\nabla \] is the notation for the gradient, $f$ is the function and $x$ is the variable.

Complete step by step solution:
Gradient is defined as the rate of change of a variable quantity with respect to distance. It is denoted by sign \[\nabla \]. So, we can say that if the units of the numerator and the denominator of the gradient function are the same then, the gradient function will be unitless.

Now, going through the given options one-by-one as-
1. Pressure gradient,
\[\nabla P = \dfrac{{dP}}{{dx}}\left( {{\text{Pascal/metre}}} \right)\]
Here, the unit of pressure is Pascal, and the distance is metre.

2. Displacement gradient,
\[\nabla X = \dfrac{{dX}}{{dx}}\,\,\left( {{\text{metre/metre}}} \right)\,\text{or unitless}\]
As the unit of distance as well as the displacement are similar, we can cancel them accordingly and ultimately make the displacement gradient as unitless.

3. Force gradient,
\[\nabla F = \dfrac{{dF}}{{dx}}\,\,\left( {{\text{Newton/metre}}} \right)\]
Here, the unit of force is Newton, and distance is metre.

4. Velocity gradient,
\[\nabla v = \dfrac{{dv}}{{dx}}\,\,\left( {{\text{metre/sec - metre}}} \right)\,\,\text{or,}\,{\text{se}}{{\text{c}}^{ - 1}}\]
Here, the unit of velocity is metre per seconds, and distance is metre.
Hence, we can say that displacement gradient is unitless quantity.

So, option b is the correct answer.

Note:To solve these types of questions, candidates must be aware about the different units of expressing the mathematical quantities as well as the gradient expression.