STATEMENT - 1 : A metallic rod placed upon smooth surface is heated. The strain produced is ZERO.
STATEMENT - 2 : Strain is non-zero only when stress is developed in the rod.
STATEMENT - 3 : Two metallic rods of length `l_(1)` and `l_(2)` and coefficient of linear expansion `alpha_(1)` and `alpha_(2)` are heated such that the difference of their length ramains same at ALL termperatures Then `(alpha_(1))/(alpha_(2))=(l_(2))/(l_(1))`

Options

A. TTT
B. FTF
C. FFT
D. FFF

Answer

To analyze the statements provided in the question, we will evaluate each statement step by step.

### Step 1: Evaluate Statement 1
**Statement 1:** A metallic rod placed upon a smooth surface is heated. The strain produced is ZERO.

- When a metallic rod is heated, it expands. However, if the rod is placed on a smooth surface and is free to expand without any constraints (i.e., it is not clamped or fixed), there will be no stress developed in the rod.
- Since stress is defined as force per unit area and there is no external force acting on the rod due to its free expansion, the stress remains zero.
- According to the relationship between stress and strain (Stress = Young's Modulus × Strain), if stress is zero, then strain must also be zero.

**Conclusion:** Statement 1 is TRUE.

### Step 2: Evaluate Statement 2
**Statement 2:** Strain is non-zero only when stress is developed in the rod.

- Strain is defined as the deformation per unit length of a material. It is a measure of how much a material deforms under stress.
- The relationship between stress and strain is given by the equation: Stress = Young's Modulus × Strain.
- This means that for strain to be non-zero, there must be some stress acting on the material. If there is no stress, there cannot be any strain.

**Conclusion:** Statement 2 is TRUE.

### Step 3: Evaluate Statement 3
**Statement 3:** Two metallic rods of length \( l_1 \) and \( l_2 \) and coefficients of linear expansion \( \alpha_1 \) and \( \alpha_2 \) are heated such that the difference of their lengths remains the same at ALL temperatures. Then \( \frac{\alpha_1}{\alpha_2} = \frac{l_2}{l_1} \).

- When the rods are heated, their lengths will change according to the formula:
\[ \Delta L_1 = \alpha_1 l_1 \Delta T \]
\[ \Delta L_2 = \alpha_2 l_2 \Delta T \]
- The problem states that the difference in lengths remains constant. Therefore:
\[ \Delta L_2 - \Delta L_1 = \text{constant} \]
- This implies:
\[ \alpha_2 l_2 \Delta T - \alpha_1 l_1 \Delta T = \text{constant} \]
- Dividing through by \( \Delta T \) (assuming \( \Delta T \) is not zero), we have:
\[ \alpha_2 l_2 - \alpha_1 l_1 = 0 \]
or
\[ \alpha_1 l_1 = \alpha_2 l_2 \]
- Rearranging gives us:
\[ \frac{\alpha_1}{\alpha_2} = \frac{l_2}{l_1} \]

**Conclusion:** Statement 3 is TRUE.

### Final Conclusion
All three statements are TRUE.

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