Find the dimensions of stress, strain and modulus of elasticity.

Explanation

To find the dimensions of stress, strain, and modulus of elasticity, we will follow these steps:

### Step 1: Determine the dimensions of stress
Stress is defined as the force applied per unit area. The formula for stress (σ) is given by:

\[ \sigma = \frac{F}{A} \]

Where:
- \(F\) is the force
- \(A\) is the area

The dimensions of force (F) can be expressed using Newton's second law, \(F = m \cdot a\), where \(m\) is mass and \(a\) is acceleration. The dimensions of mass (m) are \([M]\), and the dimensions of acceleration (a) are \([L T^{-2}]\). Therefore, the dimensions of force are:

\[ [F] = [M][L T^{-2}] = [M L T^{-2}] \]

The dimensions of area (A) are given by:

\[ [A] = [L^2] \]

Now, substituting these into the stress formula:

\[ [\sigma] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

### Step 2: Determine the dimensions of strain
Strain is defined as the change in length divided by the original length. The formula for strain (ε) is given by:

\[ \epsilon = \frac{\Delta L}{L} \]

Where:
- \(\Delta L\) is the change in length
- \(L\) is the original length

Both \(\Delta L\) and \(L\) have the same dimensions of length, which is \([L]\). Therefore, strain is dimensionless:

\[ [\epsilon] = \frac{[L]}{[L]} = 1 \]

### Step 3: Determine the dimensions of modulus of elasticity
The modulus of elasticity (E) is defined as the ratio of stress to strain. The formula for modulus of elasticity is given by:

\[ E = \frac{\sigma}{\epsilon} \]

Substituting the dimensions we found for stress and strain:

\[ [E] = \frac{[\sigma]}{[\epsilon]} = \frac{[M L^{-1} T^{-2}]}{1} = [M L^{-1} T^{-2}] \]

### Final Results
- Dimensions of Stress: \([M L^{-1} T^{-2}]\)
- Dimensions of Strain: Dimensionless (1)
- Dimensions of Modulus of Elasticity: \([M L^{-1} T^{-2}]\)