Two wires of the same length and material but different radii `r_(1)` and `r_(2)` are suspended from a rigid support both carry the same load at the lower end. The ratio of the stress devedoped in the second wire to that developed in the first wire is

Options

A. `(r_(1))/(r_(2))`
B. `(r_(1)^(2))/(r_(2)^(2))`
C. `((r_(1))/(r_(2)))^(5//2)`
D. `((r_(1))/(r_(2)))^(1//2)`

Answer

To solve the problem, we need to find the ratio of the stress developed in the second wire to that developed in the first wire. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understanding Stress**:
Stress (σ) is defined as the force (F) applied per unit area (A). Mathematically, it can be expressed as:
\[ \sigma = \frac{F}{A} \]

2. **Identify the Forces**:
Both wires are carrying the same load (W). Therefore, the force acting on both wires is the same:
\[ F_1 = F_2 = W \]

3. **Cross-Sectional Area**:
The cross-sectional area (A) of a wire with radius \( r \) is given by:
\[ A = \pi r^2 \]
Thus, for wire 1 (with radius \( r_1 \)):
\[ A_1 = \pi r_1^2 \]
For wire 2 (with radius \( r_2 \)):
\[ A_2 = \pi r_2^2 \]

4. **Calculate Stress for Each Wire**:
Using the formula for stress, we can write the stress for wire 1 (σ₁) and wire 2 (σ₂):
\[ \sigma_1 = \frac{W}{A_1} = \frac{W}{\pi r_1^2} \]
\[ \sigma_2 = \frac{W}{A_2} = \frac{W}{\pi r_2^2} \]

5. **Finding the Ratio of Stresses**:
To find the ratio of the stress in wire 2 to the stress in wire 1, we can set up the following equation:
\[ \frac{\sigma_2}{\sigma_1} = \frac{\frac{W}{\pi r_2^2}}{\frac{W}{\pi r_1^2}} = \frac{r_1^2}{r_2^2} \]

6. **Final Result**:
Therefore, the ratio of the stress developed in the second wire to that developed in the first wire is:
\[ \frac{\sigma_2}{\sigma_1} = \frac{r_1^2}{r_2^2} \]

### Conclusion:
The ratio of the stress developed in the second wire to that developed in the first wire is \(\frac{r_1^2}{r_2^2}\).

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