The dimensional formula for strain energy density is

Options

A. `M^(1)L^(2)T^(-3)`
B. `M^(1)L^(2)T^(3)`
C. `M^(1)L^(-1)T^(-2)`
D. `M^(1)L^(2)T^(-2)`

Answer

To find the dimensional formula for strain energy density, we can follow these steps:

### Step 1: Understand the Concept of Strain Energy Density
Strain energy density is defined as the energy stored in a material per unit volume due to deformation.

### Step 2: Write the Formula for Strain Energy Density
The formula for strain energy density (U) is given by:
\[ U = \frac{\text{Energy}}{\text{Volume}} \]

### Step 3: Determine the Dimensional Formula for Energy
The dimensional formula for energy (E) is:
\[ [E] = [M][L^2][T^{-2}] \]
where:
- \( M \) is mass,
- \( L \) is length,
- \( T \) is time.

### Step 4: Determine the Dimensional Formula for Volume
The dimensional formula for volume (V) is:
\[ [V] = [L^3] \]

### Step 5: Calculate the Dimensional Formula for Strain Energy Density
Now, substituting the dimensional formulas of energy and volume into the formula for strain energy density:
\[ [U] = \frac{[E]}{[V]} = \frac{[M][L^2][T^{-2}]}{[L^3]} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ [U] = [M][L^2][T^{-2}] \cdot [L^{-3}] = [M][L^{2-3}][T^{-2}] = [M][L^{-1}][T^{-2}] \]

### Step 7: Write the Final Dimensional Formula
Thus, the dimensional formula for strain energy density is:
\[ [U] = [M^1][L^{-1}][T^{-2}] \]

### Step 8: Identify the Correct Option
From the options provided:
- Option C: \( m^1 L^{-1} T^{-2} \) is the correct answer.

### Final Answer:
The dimensional formula for strain energy density is \( m^1 L^{-1} T^{-2} \).

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