Answer
Verified
399.9k+ views
Hint: Here, we will proceed by writing down the formula of strain energy. Then, by using this formula we will write the formula for strain energy density. Finally, we will apply dimensional analysis on both sides of the formula.
Step By Step Answer:
Formula Used- ${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon $.
According to strain energy formula, we can write (provided the stress is directly proportional to the strain)
${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon {\text{ }} \to {\text{(1)}}$
where U is the strain energy, V is the volume of the body, $\sigma $ denotes the stress and $\varepsilon $ is the strain
Strain energy density is defined as the strain energy per unit volume of the body
i.e., Strain energy density = $\dfrac{{{\text{Strain Energy}}}}{{{\text{Volume}}}} = \dfrac{{\text{U}}}{{\text{V}}}$
By taking volume of the body V from the RHS to the LHS of equation (1), we get
Strain energy density $\dfrac{{\text{U}}}{{\text{V}}} = \dfrac{1}{2}\sigma \varepsilon {\text{ }} \to {\text{(2)}}$
Stress is defined as the force applied or experienced per unit area
i.e., Stress $\sigma = \dfrac{{\text{F}}}{{\text{A}}}{\text{ }} \to {\text{(3)}}$ where F denotes the force applied or experienced and A denotes the area on which it is applied
As, dimensional formula for force is $\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]$ and that for area is $\left[ {{{\text{L}}^2}} \right]$
By applying dimensional analysis on both sides of equation (3), we get
Dimensional formula for stress =
\[\dfrac{{{\text{Dimensional formula for force}}}}{{{\text{Dimensional formula for area}}}} = \dfrac{{\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]}}{{\left[ {{{\text{L}}^2}} \right]}} = \left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\left[ {{{\text{L}}^{ - 2}}} \right] = \left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]
Also we know that strain is the ratio of the deformation produced after the application of a force on the body to the original dimension of the body
i.e., Strain = $\dfrac{{{\text{Deformation in dimension}}}}{{{\text{Original dimension}}}} = \dfrac{{\Delta {\text{L}}}}{{\text{L}}}$ where $\Delta {\text{L}}$denotes the change in length of the body and L denotes the original length of the body
Clearly, we can see from the formula of strain that it is a dimensionless quantity i.e., $\varepsilon $ is dimensionless
By applying dimensional analysis to equation (2), we get
Dimensional formula for stress energy density = (Dimensional formula for stress)(Dimensional formula for strain)
But since strain is dimensionless so we can write,
Dimensional formula for stress energy density = Dimensional formula for stress
$ \Rightarrow $ Dimensional formula for stress energy density = \[\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right] = \left[ {{{\text{M}}^1}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]
Therefore, option C is correct.
Note: In this particular problem, when dimensional analysis is applied to the formula i.e., Stress energy density = $\dfrac{1}{2}\sigma \varepsilon $, $\dfrac{1}{2}$ is a number (constant) and the numbers are dimensionless so its dimension will automatically be neglected. That’s why the dimension of stress energy density is equal to the product of the dimensions of stress and strain.
Step By Step Answer:
Formula Used- ${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon $.
According to strain energy formula, we can write (provided the stress is directly proportional to the strain)
${\text{U}} = \dfrac{1}{2}{\text{V}}\sigma \varepsilon {\text{ }} \to {\text{(1)}}$
where U is the strain energy, V is the volume of the body, $\sigma $ denotes the stress and $\varepsilon $ is the strain
Strain energy density is defined as the strain energy per unit volume of the body
i.e., Strain energy density = $\dfrac{{{\text{Strain Energy}}}}{{{\text{Volume}}}} = \dfrac{{\text{U}}}{{\text{V}}}$
By taking volume of the body V from the RHS to the LHS of equation (1), we get
Strain energy density $\dfrac{{\text{U}}}{{\text{V}}} = \dfrac{1}{2}\sigma \varepsilon {\text{ }} \to {\text{(2)}}$
Stress is defined as the force applied or experienced per unit area
i.e., Stress $\sigma = \dfrac{{\text{F}}}{{\text{A}}}{\text{ }} \to {\text{(3)}}$ where F denotes the force applied or experienced and A denotes the area on which it is applied
As, dimensional formula for force is $\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]$ and that for area is $\left[ {{{\text{L}}^2}} \right]$
By applying dimensional analysis on both sides of equation (3), we get
Dimensional formula for stress =
\[\dfrac{{{\text{Dimensional formula for force}}}}{{{\text{Dimensional formula for area}}}} = \dfrac{{\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]}}{{\left[ {{{\text{L}}^2}} \right]}} = \left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\left[ {{{\text{L}}^{ - 2}}} \right] = \left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]
Also we know that strain is the ratio of the deformation produced after the application of a force on the body to the original dimension of the body
i.e., Strain = $\dfrac{{{\text{Deformation in dimension}}}}{{{\text{Original dimension}}}} = \dfrac{{\Delta {\text{L}}}}{{\text{L}}}$ where $\Delta {\text{L}}$denotes the change in length of the body and L denotes the original length of the body
Clearly, we can see from the formula of strain that it is a dimensionless quantity i.e., $\varepsilon $ is dimensionless
By applying dimensional analysis to equation (2), we get
Dimensional formula for stress energy density = (Dimensional formula for stress)(Dimensional formula for strain)
But since strain is dimensionless so we can write,
Dimensional formula for stress energy density = Dimensional formula for stress
$ \Rightarrow $ Dimensional formula for stress energy density = \[\left[ {{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right] = \left[ {{{\text{M}}^1}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}} \right]\]
Therefore, option C is correct.
Note: In this particular problem, when dimensional analysis is applied to the formula i.e., Stress energy density = $\dfrac{1}{2}\sigma \varepsilon $, $\dfrac{1}{2}$ is a number (constant) and the numbers are dimensionless so its dimension will automatically be neglected. That’s why the dimension of stress energy density is equal to the product of the dimensions of stress and strain.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE