Question

Which of the following is the dimension of compressibility?
A). $[{{M}^{-1}}{{L}^{1}}{{T}^{2}}]$
B). $[{{M}^{1}}{{L}^{-1}}{{T}^{-2}}]$
C). $[{{M}^{-1}}{{L}^{-1}}{{T}^{-2}}]$
D). $[{{M}^{-1}}{{L}^{-1}}{{T}^{2}}]$

Answer 1

Hint: We need to solve this question with reference to compressibility. It is the relation between the tangential stress and the shearing strain of a particular material. To determine the dimension of compressibility, we first need to know the dimensions of the tangential stress and the strain that are the fundamental building blocks of compressibility. With this information, we can finally equate the unit of compressibility and find its dimension. We can also derive its dimension directly by knowing it’s SI unit. This approach will help us to solve the given question easily.

Complete step by step answer:
Compressibility is defined as the ratio between tangential stress to the shearing strain, However, it should be within the elastic limit. The unit of compressibility is $N/{{m}^{2}}$ or also known as Pascal.
So, from the unit we can derive the dimensional formula. The dimensional formula of N or newton or force is mass $\times$acceleration or [$ML{{T}^{-2}}$]. And by ${{m}^{2}}$the dimension becomes [${{L}^{2}}$].
So, the dimension of compressibility is $[\dfrac{ML{{T}^{-2}}}{{{L}^{2}}}]=[M{{L}^{-1}}{{T}^{-2}}]$.
Hence the dimension of compressibility is $[{{M}^{1}}{{L}^{-1}}{{T}^{-2}}]$.

So, the correct answer is Option B.

Note: We must keep in mind that dimensional formula is basically an expression for units of any physical quantity in the terms of fundamental quantities. The fundamental quantities are coined as mass (M), length (L), and time (T). The formula for any SI unit is expressed in the terms of the powers of mass, length, and time. It is used because it shows a relation between how and which of the fundamental quantities represent the dimensions of a physical quantity. It gives us an idea of the quantities that might be used to derive a given quantity.