An iron of length l and having cross-section A is heated form `0^(@)C` to `100^(@)`C. IF his bar is held so that it is not permitted to expand or bend, the gigantic force that is developed is

Options

A. directly proportional to the length of the bar
B. inversely proportional to the length of the bar
C. independent of the length of the bar
D. inversely proportional too the cross-section of the bar

Answer

To solve the problem of the force developed in an iron bar when it is heated from 0°C to 100°C while being held fixed, we can follow these steps:

### Step 1: Understand the Concept of Thermal Expansion
When a material is heated, it tends to expand. The amount of expansion can be quantified using the coefficient of linear expansion (α). The change in length (ΔL) due to a temperature change (Δθ) can be expressed as:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta \theta \]
where \( L_0 \) is the original length of the bar.

### Step 2: Calculate the Change in Length
In this case, the original length \( L_0 \) is \( L \), and the temperature change \( \Delta \theta \) is from 0°C to 100°C, which is 100°C. Therefore:
\[ \Delta L = L \cdot \alpha \cdot 100 \]

### Step 3: Define Stress and Strain
Stress (σ) is defined as force (F) per unit area (A):
\[ \sigma = \frac{F}{A} \]
Strain (ε) is defined as the change in length per original length:
\[ \epsilon = \frac{\Delta L}{L} \]

### Step 4: Relate Stress and Strain
According to Hooke's Law, stress is proportional to strain:
\[ \sigma = Y \cdot \epsilon \]
where \( Y \) is the Young's modulus of the material.

### Step 5: Substitute Strain into the Stress Equation
Substituting the expression for strain into the stress equation gives:
\[ \sigma = Y \cdot \frac{\Delta L}{L} \]
Substituting ΔL from Step 2:
\[ \sigma = Y \cdot \frac{L \cdot \alpha \cdot 100}{L} \]
The \( L \) cancels out:
\[ \sigma = Y \cdot \alpha \cdot 100 \]

### Step 6: Relate Stress to Force
Now, substituting the expression for stress back into the force equation:
\[ \frac{F}{A} = Y \cdot \alpha \cdot 100 \]
Rearranging gives:
\[ F = A \cdot Y \cdot \alpha \cdot 100 \]

### Step 7: Analyze the Result
From the equation \( F = A \cdot Y \cdot \alpha \cdot 100 \), we can see that:
- The force \( F \) is directly proportional to the cross-sectional area \( A \).
- The force \( F \) is independent of the length \( L \).

### Conclusion
The gigantic force developed when the iron bar is heated is independent of its length.