Question

Give examples of three physical quantities that are having dimensions [ $M{L}^{-1}{T}^{-2}$ ].

Answer 1

Pressure

1. Pressure is the amount of force acting on a unit area. We can write the expression for pressure as, $P=\frac{F}{A}$
2. Where $P$ stands for the pressure, $F$ stands for the force acting on a surface, and $A$ stands for the area of the surface.
3. Force is the product of acceleration and mass. Hence we can write the dimensional formula for force as, $F=\left[ML{T}^{-2}\right]$
4. The dimensional formula for the area can be written as, $A=\left[L^2\right]$
5. Substituting the dimensional formulae for force and area in the expression of pressure, we get $P=\frac{F}{A}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

Stress

1. Stress is the external restoring force that acts per unit area. We can write the expression for pressure as,$\sigma =\frac{F}{A}$ Where $\sigma$ stands for the stress, $F$ stands for the force acting on the object, and $A$ stands for the cross-sectional area.
2. Force is the product of acceleration and mass. Hence we can write the dimensional formula for force as, $F=\left[ML{T}^{-2}\right]$
3. The dimensional formula for the area can be written as, $A=\left[L^2\right]$
4. Substituting the dimensional formulae for force and area in the expression of stress, we get $\sigma =\frac{F}{A}=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

Coefficient of elasticity

1. The internal restoring force acting per unit area of the cross-section of the deformed body is called the coefficient of elasticity.
2. Stress is the external restoring force that acts per unit area.
3. As stress is directly proportional to strain, therefore we can say that stress by strain leads to the constant term. Therefore, $coefficientofelasticity=\frac{stress}{strain}$
4. As we have discussed above the dimensional formula of stress = $\left[M{L}^{-1}{T}^{-2}\right]$
5. A detailed description and prediction of the behavior of elastic, plastic, and fluid materials are made possible by stress, which is defined as the force per unit area that exists within materials as a result of externally applied forces, unequal heating, or persistent deformation. Because strain is the ratio of identical physical qualities in the same dimension, it has no dimensions. $Strain=\Delta L\times L^{-1}=[M^0{L}^0{T}^0]$
6. Substituting the dimensional formulae for stress and strain in the expression coefficient of elasticity, we get $coefficientofelasticity=\frac{stress}{strain}=\frac{M{L}^{-1}{T}^{-2}}{M^0{L}^0{T}^0}=[M{L}^{-1}{T}^{-2}]$

Hence, pressure, stress, and coefficient of elasticity are the three examples of physical quantities with dimensions [ $M{L}^{-1}{T}^{-2}$ ].