Question

The given dimensional formula is for: $M\text{ }{{L}^{2}}\text{ }{{T}^{-2}}$
(A) Moment of inertia
(B) Pressure
(C) Elasticity
(D) Couple acting on a body

Answer 1

Hint : A dimensional formula is dependent on the quantities it signifies. To attain the dimensions, you must know their respective units. In the dimensional formula, M stands for the mass of the body, L stands for the length, and T stands for the time.

Complete Step By Step Answer:
To find the correct answer, we have to first find the Dimensional formula for each of the given options. Let’s see each Option in detail.
Option A: Moment of Inertia
It is defined as the angular acceleration resisted by the body that is the product of the mass of the body and the square of the distance between them.
$Moment\text{ }of\text{ }Inertia\text{ }=\text{ }m\text{ }\times \text{ }{{\text{r}}^{2}}$
The dimensional unit of m is ${{M}^{1}}$ and that of ${{r}^{2}}$ is ${{L}^{2}}$ .
Thus dimensional formula of Moment of Inertia is ${{M}^{1}}\text{ }{{L}^{2}}\text{ }{{\text{T}}^{0}}$
Therefore, Option A is incorrect.
Option B: Pressure
Pressure is expressed as the product of force and area. Force is the product of mass and acceleration.
$\begin{aligned} & \therefore \text{ }Pressure\text{ = Force }\times \text{ Area} \\\ & \text{ = mass }\times \text{ acceleration }\times \text{ area} \\\ \end{aligned}$
The dimensional unit of m is ${{M}^{1}}$ , acceleration is $L\text{ }{{\text{T}}^{-2}}$ and the area is ${{L}^{2}}$ .
Thus dimensional formula of Pressure is ${{M}^{1}}\text{ }{{L}^{-1}}\text{ }{{\text{T}}^{-2}}$
Therefore, Option B is incorrect.
Option C: Elasticity
Elasticity is expressed as the ratio of stress and strain.
$Elasticity\text{ = }\dfrac{Stress}{Strain}$
$Stress\text{ = Force }\times \text{ }{{\left( \text{area} \right)}^{-1}}$
The dimensions of force are ${{M}^{1}}\text{ }{{L}^{1}}\text{ }{{\text{T}}^{-2}}$ and the area is ${{L}^{2}}$ . Thus, the dimensional formula of Stress is ${{M}^{1}}\text{ }{{L}^{-1}}\text{ }{{\text{T}}^{-2}}$
Similarly, the dimensional formula for Strain is ${{M}^{0}}\text{ }{{L}^{0}}\text{ }{{\text{T}}^{0}}$ .
Therefore, the dimensional formula of Elasticity ${{M}^{1}}\text{ }{{L}^{-1}}\text{ }{{\text{T}}^{-2}}$ .
Therefore, Option C is incorrect.
Option D: Couple acting on a body
It is also known as Torque. It is expressed as the product of angular acceleration and Moment of inertia. We have seen the dimensions of Moment of Inertia in Option A.
$\therefore \text{ }Moment\text{ }of\text{ }Inertia\text{ }=\text{ }{{M}^{1}}\text{ }{{L}^{2}}\text{ }{{\text{T}}^{0}}$
Dimensions of angular acceleration are ${{M}^{0}}\text{ }{{L}^{0}}\text{ }{{\text{T}}^{-2}}$
Thus, the dimensional formula of a Couple acting on a body is $M\text{ }{{L}^{2}}\text{ }{{T}^{-2}}$ .
Therefore, Option D is correct.
Final answer: The correct answer is Option D: Couple acting on a body.

Note :
To find the dimensional formula of any quantity you must know the units they are expressed in and the representation of the fundamental quantities in the dimensional formula. Also, you must know the formula of the respective quantity. Because if you cannot recall the dimensional formula of the asked quantity, you can derive it using the formula.