A rigid bar of mass M is supported symmetrically by three wires each of length l. Those at each end are of copper and the middle one is of iron. The ratio of their diameters, if each is to have the same tension, is equal to

Options

A. `(sqrt(Y_("copper")))/(Y_("iron"))`
B. `sqrt((Y_("iron"))/(Y_("copper")))`
C. `(Y_("iron")^(2))/(Y_("copper")^(2))`
D. `(Y_("iron"))/(Y_("copper"))`

Explanation

To solve the problem, we need to determine the ratio of the diameters of the copper and iron wires that support a rigid bar symmetrically, given that each wire experiences the same tension.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress to strain. Stress is the force per unit area, and strain is the change in length divided by the original length.

2. **Expression for Stress**:
For a wire, stress can be expressed as:
\[ \text{Stress} = \frac{F}{A} \]
where \( F \) is the force (tension) and \( A \) is the cross-sectional area. The area \( A \) of a circular wire can be expressed in terms of its diameter \( d \):
\[ A = \frac{\pi d^2}{4} \]
Therefore, the stress in terms of diameter becomes:
\[ \text{Stress} = \frac{4F}{\pi d^2} \]

3. **Relating Stress to Strain**:
From Young's modulus, we have:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{4F}{\pi d^2} \cdot \frac{L_0}{\Delta L} \]
Rearranging gives:
\[ d^2 = \frac{4F L_0}{\pi Y \Delta L} \]

4. **Considering the Wires**:
Since the tension (force) in each wire is the same and the original length \( L_0 \) is the same for all wires, we can denote:
- For copper: \( d_c^2 = \frac{4F L_0}{\pi Y_c \Delta L} \)
- For iron: \( d_i^2 = \frac{4F L_0}{\pi Y_i \Delta L} \)

5. **Finding the Ratio of Diameters**:
Taking the ratio of the diameters of copper and iron:
\[ \frac{d_c^2}{d_i^2} = \frac{Y_i}{Y_c} \]
Therefore, taking the square root gives us:
\[ \frac{d_c}{d_i} = \sqrt{\frac{Y_i}{Y_c}} \]

6. **Conclusion**:
This means that the ratio of the diameters of the copper wire to the iron wire is equal to the square root of the ratio of their Young's moduli.

### Final Answer:
The ratio of the diameters \( \frac{d_c}{d_i} \) is given by:
\[ \frac{d_c}{d_i} = \sqrt{\frac{Y_i}{Y_c}} \]