The modulus of elasticity is dimensionally equivalent to

Options

A. Stress
B. Surface tension
C. Strain
D. Coeefficient of viscosity

Answer

To solve the question regarding the dimensional equivalence of the modulus of elasticity, we can follow these steps:

### Step 1: Understand the Definition of Modulus of Elasticity
The modulus of elasticity (E) is defined as the ratio of stress (σ) to strain (ε):
\[ E = \frac{\sigma}{\epsilon} \]

### Step 2: Identify the Dimensions of Stress and Strain
- **Stress (σ)** is defined as force per unit area. The dimension of force (F) is given by:
\[ [F] = [M][L][T^{-2}] \]
where [M] is mass, [L] is length, and [T] is time.

The area (A) has the dimension:
\[ [A] = [L^2] \]

Therefore, the dimension of stress is:
\[ [\sigma] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

- **Strain (ε)** is defined as the ratio of change in length to the original length. It is a dimensionless quantity, meaning:
\[ [\epsilon] = 1 \]

### Step 3: Substitute the Dimensions into the Modulus of Elasticity Formula
Since strain is dimensionless, we can simplify the equation for modulus of elasticity:
\[ E = \frac{\sigma}{\epsilon} = \frac{\sigma}{1} = \sigma \]

### Step 4: Conclude the Dimensional Equivalence
Thus, the modulus of elasticity has the same dimensions as stress:
\[ [E] = [\sigma] = [M][L^{-1}][T^{-2}] \]

### Final Answer
The modulus of elasticity is dimensionally equivalent to stress.

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