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Question: Which of the following cannot be expressed as \(dyne\,c{m^{ - 2}}\)? A) Pressure B) Longitudinal...

Which of the following cannot be expressed as dyne cm−2dyne\,c{m^{ - 2}}?
A) Pressure
B) Longitudinal Stress
C) Longitudinal Strain
D) Young’s Modulus of Elasticity

Explanation

Solution

After looking at the dimensional formula of dyne cm−2dyne\,c{m^{ - 2}} , the question will get easier. We can easily get the dimensional formula of the quantities mentioned in the options. After matching these dimensional formulas with that of dyne cm−2dyne\,c{m^{ - 2}} , we can find the answer.

Formula Used:
Dimensional formula for dynecm2=[M1L1T−2][L2]=[M1L−1T−2]\dfrac{{dyne}}{{c{m^2}}} = \dfrac{{[{M^1}{L^1}{T^{ - 2}}]}}{{[{L^2}]}} = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Dimensional formula for Pressure =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Dimensional formula for Longitudinal stress =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Dimensional formula for Longitudinal Strain =[M0L0T0] = [{M^0}{L^0}{T^0}]
Dimensional formula for Young’s Modulus =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]

Complete step by step solution:
First of all, we will look at the dimensional formula for given unit dyne cm−2dyne\,c{m^{ - 2}}
Dimensional formula for dynecm2=[M1L1T−2][L2]=[M1L−1T−2]\dfrac{{dyne}}{{c{m^2}}} = \dfrac{{[{M^1}{L^1}{T^{ - 2}}]}}{{[{L^2}]}} = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Now that we have the dimensional formula for the given unit, we will look at the dimensional formulas of quantities given in the option and check if they match. If they do, that particular quantity can be expressed as dyne cm−2dyne\,c{m^{ - 2}} otherwise not.
Option A: Pressure
Dimensional formula for Pressure =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Its dimensional formula is exactly similar to that of the given unit. Therefore, it can be expressed as dyne cm−2dyne\,c{m^{ - 2}}

Option B: Longitudinal stress
Dimensional formula for Longitudinal stress =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Its dimensional formula is exactly similar to that of the given unit. Therefore, it can be expressed as dyne cm−2dyne\,c{m^{ - 2}}

Option C: Longitudinal Strain
Dimensional formula for Longitudinal Strain =[M0L0T0] = [{M^0}{L^0}{T^0}]
Its dimensional formula is not similar to that of the given unit. Therefore, it cannot be expressed as dyne cm−2dyne\,c{m^{ - 2}}

Option D: Young;s Modulus of Elasticity
Dimensional formula for Young’s Modulus =[M1L−1T−2] = [{M^1}{L^{ - 1}}{T^{ - 2}}]
Its dimensional formula is exactly similar to that of the given unit. Therefore, it can be expressed as dyne cm−2dyne\,c{m^{ - 2}}

Therefore, Option C is correct.

Note: Dimensional formulas make it easier to identify a quantity’s unit. It depicts the basic dimensions of a quantity. Sometimes, the units of two quantities are different. So, we can check their dimensional formulas to figure if they are the same quantity or not. However, sometimes two different quantities can also have the same dimensional formula.