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Question: The dimensional formula of modulus of elasticity is A) \( [M{L^{ - 1}}{T^{ - 2}}] \) B) \( [{M^...

The dimensional formula of modulus of elasticity is
A) [ML−1T−2][M{L^{ - 1}}{T^{ - 2}}]
B) [M0LT−2][{M^0}L{T^{ - 2}}]
C) [MLT−2][ML{T^{ - 2}}]
D) [ML2T−2][M{L^2}{T^{ - 2}}]

Explanation

Solution

Hint : The modulus of elasticity of any object is the ratio of the stress acting on the object to the strain it experiences. We will find the dimensional formula of stress and strain and then use their ratio to find the dimensional formula for modulus of elasticity.

Formula Used: In this solution we will be using the following formula,
⇒σ = FA\Rightarrow \sigma {\text{ = }}\dfrac{F}{A} where σ\sigma is the stress, FF is the force and AA is the area of cross-section.

Complete step by step answer
We know that the modulus of elasticity (Y)(Y) of an object is given as the ratio of the stress to the strain of the body. Let us start by finding the dimensional formula of stress.
We know that stress is defined as the ratio of force and area i.e.
Stress σ = FA\sigma {\text{ = }}\dfrac{F}{A}
Since force has the dimensions [F]=[m][a]=M1L1T−2[F] = [m][a] = {M^1}{L^1}{T^{ - 2}} and
Area has the dimensions [A]=L2[A] = {L^2} , stress will have the dimensional formula
⇒[σ]=M1L1T−2L2\Rightarrow [\sigma ] = \dfrac{{{M^1}{L^1}{T^{ - 2}}}}{{{L^2}}}
⇒[σ]=M1L−1T−2\Rightarrow [\sigma ] = {M^1}{L^{ - 1}}{T^{ - 2}}
Now let’s find the dimensional formula of strain experienced by the object. Strain (E)(E) is defined as the ratio of the change in length of the object to its original length when an external force is applied to the object. Since both of these quantities have the dimensions of length, their ratio will be dimensionless i.e.
⇒[E]=M0L0T0\Rightarrow [E] = {M^0}{L^0}{T^0}
Then the dimensional formula of modulus of elasticity will be
⇒[Y]=[σ][E] ⇒[Y]=M1L−1T−2M0L0T0 \begin{gathered} \Rightarrow [Y] = \dfrac{{[\sigma ]}}{{[E]}} \\\ \Rightarrow [Y] = \dfrac{{{M^1}{L^{ - 1}}{T^{ - 2}}}}{{{M^0}{L^0}{T^0}}} \\\ \end{gathered}
Which on simplifying gives us
⇒[Y]=M1L−1T−2\Rightarrow [Y] = {M^1}{L^{ - 1}}{T^{ - 2}} which is the dimensional formula given in option (A) that is correct.

Note
We don’t need to think about the different kinds of stresses that can act on an object since all the kinds of stresses will have the same dimensional formula. Also, since the strain experienced by an object is always dimensionless, the modulus of elasticity will have the same dimensions as stress itself.