What is the dimensional formula of tensile stress?

Answer

To find the dimensional formula of tensile stress, we can follow these steps:

### Step 1: Understand the Definition of Tensile Stress
Tensile stress (σ) is defined as the force (F) applied per unit area (A) of a material. Mathematically, it is expressed as:
\[ \sigma = \frac{F}{A} \]

### Step 2: Identify the Units of Force and Area
- The unit of force (F) is Newton (N).
- The unit of area (A) is square meters (m²).

### Step 3: Find the Dimensional Formula of Force
Force can be expressed using Newton's second law, which states that force is the product of mass (m) and acceleration (a):
\[ F = m \cdot a \]
The dimensional formula for mass (m) is:
\[ [m] = M \]
The dimensional formula for acceleration (a) is:
\[ [a] = \frac{[L]}{[T^2]} = LT^{-2} \]
Thus, the dimensional formula for force (F) is:
\[ [F] = [m] \cdot [a] = M \cdot LT^{-2} = MLT^{-2} \]

### Step 4: Find the Dimensional Formula of Area
Area is defined as the product of length and width. Therefore, the dimensional formula for area (A) is:
\[ [A] = [L^2] \]

### Step 5: Combine the Dimensional Formulas to Find Tensile Stress
Now we can substitute the dimensional formulas of force and area into the equation for tensile stress:
\[ [\sigma] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} \]

### Step 6: Simplify the Expression
When we simplify the expression:
\[ [\sigma] = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Final Result
Thus, the dimensional formula for tensile stress is:
\[ [\sigma] = ML^{-1}T^{-2} \]

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