A rectangular beam of metal supported at its two ends is loaded at the centre. The depression at the centre is proportional to

Options

A. `(1)/(Y)`
B. `Y^(2)`
C. `(1)/(Y^(2))`
D. Y

Explanation

To solve the problem, we need to analyze the relationship between the depression (deflection) of a rectangular beam supported at its ends and the parameters involved, particularly focusing on Young's modulus of elasticity.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A rectangular beam is supported at both ends and a load (mass) is applied at the center. This setup creates a deflection (depression) at the center of the beam.

2. **Identifying the Parameters**:
- Let:
- \( L \) = length of the beam
- \( B \) = width of the beam
- \( D \) = thickness (depth) of the beam
- \( m \) = mass applied at the center
- \( g \) = acceleration due to gravity
- \( Y \) = Young's modulus of elasticity of the material

3. **Using the Deflection Formula**:
- The formula for the deflection \( \delta \) at the center of a simply supported beam under a central load is given by:
\[ \delta = \frac{mgL^3}{4BD^3Y} \]
- Here, \( mg \) is the weight of the load, and the other parameters are as defined above.

4. **Analyzing the Proportionality**:
- From the formula, we can see that the deflection \( \delta \) is directly proportional to the load \( mg \) and \( L^3 \), and inversely proportional to \( Y \) (Young's modulus), \( B \), and \( D^3 \).
- Therefore, we can express the relationship as:
\[ \delta \propto \frac{mgL^3}{YBD^3} \]

5. **Conclusion**:
- Since we are specifically interested in how the depression \( \delta \) is proportional to Young's modulus \( Y \), we can conclude that:
\[ \delta \propto \frac{1}{Y} \]
- This means that as the Young's modulus increases, the deflection decreases, indicating that stiffer materials (higher Young's modulus) will deflect less under the same load.

### Final Answer:
The depression at the center is proportional to \( \frac{1}{Y} \) (where \( Y \) is Young's modulus of elasticity).

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