Question

The modulus of elasticity is dimensionally equivalent to

• A. strain
• B. force
• C. stress
• D. coefficient of viscosity

Answer 1

The correct option is C

stress

Explanation for Correct Option:

Let $F$ be the force applied, $A$ be the area, $△x$ be the change in dimension and $x$ be the original dimension.

We know that the modulus of elasticity is given dimensionally as

$$\begin{gathered} \lambda =\frac{stress}{strain} \\ =\frac{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$A$}\right.}}{{\displaystyle \raisebox{1ex}{$△x$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}} \\ =\frac{{\displaystyle \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$m^2$}\right.}}{{\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}} \\ =\frac{N}{m^2} \\ =N{m}^{-2} \end{gathered}$$

Hence, the dimension of modulus of elasticity is $N{m}^{-2}$.

Option (C):

We know that stress can be dimensionally given as,

$$\begin{gathered} stress=\frac{F}{A} \\ =\frac{N}{m^2} \\ =N{m}^{-2} \end{gathered}$$

Hence, the dimension of stress is $N{m}^{-2}$ which is also equivalent to the modulus of elasticity.

Explanation for Incorrect Options:

Option (A):

We know that strain can be dimensionally given as,

$$\begin{gathered} strain=\frac{△x}{x} \\ =\frac{m}{m} \\ =1 \end{gathered}$$

Hence, the strain is a dimensionless quantity and is therefore not equivalent to the modulus of elasticity.

Option (B):

The SI unit of force is just Newton or $N$.

Hence, the quantity force is not equivalent to the modulus of elasticity.

Option (D):

Let $v$ be the velocity and $x$ be the displacement.

The coefficient of viscosity is given as,

$$\begin{gathered} \eta =\frac{F}{A\left({\displaystyle \raisebox{1ex}{$dv$}\!\left/ \!\raisebox{-1ex}{$dx$}\right.}\right)} \\ =\frac{N}{m^2\left({\displaystyle \raisebox{1ex}{$m{s}^{-1}$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}\right)} \\ =N{m}^{-2}s \end{gathered}$$

Hence, the coefficient of viscosity is not equivalent to the modulus of elasticity.

Thus, the modulus of elasticity is dimensionally equivalent to stress.

Hence, option (C) is the correct answer.