Consider two cylindrical rods of indentical dimesnions, one of rubber and the other of steel. Both the rods are fixed rigidiy at one end to the roof. `A` mass `M` is attached to each of the free ends at the centre of the rods.

Options

A. Both the rods will elongate but there shall be no perceptible change in shape
B. The steel rod will elongate and change shape but the rubber rod will only elongate
C. The steel rod will elongate without any perceptible change in shape, but the rubber rod will elongate and the shape of the bottom edge will change to an ellipse
D. The steel rod will elongate, without any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the centre

Explanation

To solve the problem, we need to analyze how two cylindrical rods made of different materials (rubber and steel) behave when a mass \( M \) is attached to their free ends. Both rods are fixed rigidly at one end.

### Step-by-Step Solution:

1. **Identify the Materials and Properties**:
- We have two rods: one made of rubber and the other made of steel.
- Both rods have identical dimensions (length, radius).

2. **Understand the Setup**:
- Both rods are fixed at one end to the roof.
- A mass \( M \) is attached to the free end of each rod.

3. **Consider the Effects of the Mass**:
- When the mass \( M \) is applied, both rods will experience elongation due to the tensile force.
- The elongation will be influenced by the material properties of the rods.

4. **Introduce Poisson's Ratio**:
- Poisson's ratio (\( \sigma \)) is defined as the ratio of lateral strain to longitudinal strain.
- For rubber, \( \sigma \approx 0.5 \).
- For steel, \( \sigma \approx 0.3 \).

5. **Analyze the Elongation**:
- Both rods will elongate under the applied mass, but the manner of elongation will differ due to their material properties.
- The steel rod will elongate without any perceptible change in its shape because of its lower Poisson's ratio.
- The rubber rod, due to its higher Poisson's ratio, will elongate but also change shape, tapering towards the center.

6. **Evaluate the Options**:
- **Option 1**: Incorrect, as the rubber rod will change shape.
- **Option 2**: Incorrect, as both rods will elongate, but the rubber rod will also change shape.
- **Option 3**: Incorrect, as the rubber rod will not change to an elliptical shape but will taper.
- **Option 4**: Correct, as the steel rod will elongate without perceptible change, while the rubber rod will elongate and taper at the center.

### Conclusion:
The correct answer is **Option 4**: The steel rod will elongate without any perceptible change in shape, while the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the center.