Dimensional formula for volume elasticity is

Options

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

Answer

To find the dimensional formula for volume elasticity (bulk modulus of elasticity), we can follow these steps:

### Step 1: Understand the definition of volume elasticity
Volume elasticity, also known as bulk modulus of elasticity (K), is defined as the ratio of hydraulic stress to volumetric strain.

### Step 2: Write the formula for bulk modulus of elasticity
The formula for bulk modulus of elasticity is given by:
\[ K = \frac{\text{Hydraulic Stress}}{\text{Volumetric Strain}} \]

### Step 3: Determine the dimensions of hydraulic stress
Hydraulic stress is defined as force per unit area.

- The dimension of force (F) is:
\[ [F] = MLT^{-2} \]

- The dimension of area (A) is:
\[ [A] = L^2 \]

Thus, the dimension of hydraulic stress is:
\[ \text{Hydraulic Stress} = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Step 4: Determine the dimensions of volumetric strain
Volumetric strain is defined as the change in volume divided by the original volume.

- The dimension of volume (V) is:
\[ [V] = L^3 \]

Thus, the volumetric strain is:
\[ \text{Volumetric Strain} = \frac{\Delta V}{V} = \frac{L^3}{L^3} = L^0 \]

### Step 5: Combine the dimensions to find the dimensional formula for volume elasticity
Now we can substitute the dimensions of hydraulic stress and volumetric strain into the formula for bulk modulus:
\[ K = \frac{\text{Hydraulic Stress}}{\text{Volumetric Strain}} = \frac{ML^{-1}T^{-2}}{L^0} = ML^{-1}T^{-2} \]

### Conclusion
The dimensional formula for volume elasticity (bulk modulus of elasticity) is:
\[ [K] = ML^{-1}T^{-2} \]