Four rods `A, B, C` and `D`) of the same length and material but of different radii `r, rsqrt(2),rsqrt(3)` and `2r`, respectively, are held between two rigid walls. The temperature of all rods is increased through the same range. If the rods do not bend, then

Options

A. the stress in the rods `A, B, C` and `D` is in the ratio `1:2:3:4`
B. the forces on them exerted by the wall are in the ratio `1:2:3:4`
C. the energy stored in the rods due to elasticity is in the ratio `1:2:3:4 `
D. it is independent of area like surface tension while friction depends

Explanation

To solve the problem, we need to analyze the situation involving the four rods (A, B, C, and D) of the same length and material but different radii. We will derive the relationships for stress, force, and energy stored in the rods when subjected to the same temperature change.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- All rods (A, B, C, D) have the same length (L) and material.
- The temperature change (ΔT) is the same for all rods.
- The radii of the rods are:
- Rod A: r
- Rod B: r√2
- Rod C: r√3
- Rod D: 2r

2. **Calculate the Cross-Sectional Areas**:
- The cross-sectional area (A) of a rod is given by the formula:
\[ A = \pi r^2 \]
- Therefore, the areas for each rod are:
- \( A_A = \pi r^2 \)
- \( A_B = \pi (r\sqrt{2})^2 = 2\pi r^2 \)
- \( A_C = \pi (r\sqrt{3})^2 = 3\pi r^2 \)
- \( A_D = \pi (2r)^2 = 4\pi r^2 \)

3. **Calculate the Stress in Each Rod**:
- Stress (σ) is defined as force (F) per unit area (A):
\[ \sigma = \frac{F}{A} \]
- Given that the rods do not bend, the thermal stress can also be expressed as:
\[ \sigma = \alpha \Delta T \cdot Y \]
where \( \alpha \) is the coefficient of thermal expansion and \( Y \) is Young's modulus.
- Since \( \alpha \), \( \Delta T \), and \( Y \) are the same for all rods, the stress in each rod is the same:
\[ \sigma_A = \sigma_B = \sigma_C = \sigma_D \]

4. **Calculate the Forces Exerted by the Walls**:
- Rearranging the stress formula gives us:
\[ F = \sigma \cdot A \]
- Therefore, the forces for each rod can be expressed as:
- \( F_A = \sigma \cdot A_A = \sigma \cdot \pi r^2 \)
- \( F_B = \sigma \cdot A_B = \sigma \cdot 2\pi r^2 \)
- \( F_C = \sigma \cdot A_C = \sigma \cdot 3\pi r^2 \)
- \( F_D = \sigma \cdot A_D = \sigma \cdot 4\pi r^2 \)
- The ratio of forces is:
\[ F_A : F_B : F_C : F_D = 1 : 2 : 3 : 4 \]

5. **Calculate the Energy Stored in Each Rod**:
- The energy (E) stored in a rod due to elasticity can be expressed as:
\[ E = \frac{1}{2} \cdot \text{strain}^2 \cdot A \cdot L \]
- The strain (ε) is given by:
\[ \text{strain} = \alpha \Delta T \]
- Since the strain, length (L), and Young's modulus (Y) are the same for all rods, the energy stored in each rod depends on the area:
- \( E_A = \frac{1}{2} \cdot (\alpha \Delta T)^2 \cdot A_A \cdot L \)
- \( E_B = \frac{1}{2} \cdot (\alpha \Delta T)^2 \cdot A_B \cdot L \)
- \( E_C = \frac{1}{2} \cdot (\alpha \Delta T)^2 \cdot A_C \cdot L \)
- \( E_D = \frac{1}{2} \cdot (\alpha \Delta T)^2 \cdot A_D \cdot L \)
- The ratio of energies is:
\[ E_A : E_B : E_C : E_D = 1 : 2 : 3 : 4 \]

### Conclusion:
- The stress in the rods is the same for all, hence the ratio is 1:1:1:1 (Option A is incorrect).
- The forces exerted by the walls are in the ratio 1:2:3:4 (Option B is correct).
- The energy stored in the rods is also in the ratio 1:2:3:4 (Option C is correct).
- Option D is incorrect as it does not make sense in this context.