Question

The dimensional formula of velocity gradient is .
A. $\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}} \right]$
B. \[\left[ ML{{T}^{-1}} \right]\]
C. \[\left[ M{{L}^{0}}{{T}^{-1}} \right]\]
D. \[\left[ {{M}^{0}}L{{T}^{-2}} \right]\]

Answer 1

Hint: To write dimensional formulas for any quantity we need a formula for that. Then we write formulas in basic terms like mass , time, length etc.
When we calculate velocity at per unit distance it is known as velocity gradient.

Complete step-by-step answer:
As velocity gradient can be defined as velocity divided by distance. We can also say that velocity gradient is the rate of velocity per unit distance.
We can write formula for velocity gradient as below:
$velocity\,gradient\,=\,\dfrac{velocity}{dis\tan ce}$
Now we can write dimensional formulas for each quantity.
Dimensional formula for velocity is $\left[ L{{T}^{-1}} \right]$ because velocity is defined as the rate of change of distance with time.
Dimensional formula for distance is $\left[ L \right]$
Now we can find dimensional formula for velocity gradient is
\[\Rightarrow \left[ velocity\,gradient \right]=\dfrac{\left[ L{{T}^{-1}} \right]}{\left[ L \right]}\]
\[\Rightarrow \left[ velocity\,gradient \right]=\left[ {{T}^{-1}} \right]\]
We can write it as
\[\Rightarrow \left[ velocity\,gradient \right]=\left[ {{M}^{0}}{{L}^{0}}{{T}^{-1}} \right]\]
Hence option A 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 the 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:
${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$