Ratio of energy density of two steel rods is `1 : 4` when same mass is suspended from the rods. If length of both rods is same then ratio of diameter of rods will be.

Options

A. `sqrt2:1`
B. `1:2`
C. `2:1`
D. `1:sqrt2`

Answer

To solve the problem, we need to find the ratio of the diameters of two steel rods given that the ratio of their energy densities is 1:4 when the same mass is suspended from both rods and their lengths are the same.

### Step-by-Step Solution:

1. **Understanding Energy Density**:
The energy density (U) in a material can be expressed as:
\[ U = \frac{F^2}{2A^2Y} \]
where \(F\) is the force applied (which is equal to the weight of the mass suspended), \(A\) is the cross-sectional area of the rod, and \(Y\) is Young's modulus of the material.

2. **Given Information**:
- The energy density ratio of the two rods is given as \( \frac{U_1}{U_2} = \frac{1}{4} \).
- The lengths of both rods are the same, and the same mass is suspended from both.

3. **Relating Energy Density to Area**:
Since the force \(F\) and Young's modulus \(Y\) are the same for both rods, we can simplify the ratio of energy densities:
\[ \frac{U_1}{U_2} = \frac{A_2^2}{A_1^2} \]
Therefore, we can write:
\[ \frac{A_1^2}{A_2^2} = \frac{1}{4} \]

4. **Expressing Area in Terms of Diameter**:
The cross-sectional area \(A\) of a rod can be expressed in terms of its diameter \(d\):
\[ A = \frac{\pi d^2}{4} \]
Thus, we can write:
\[ A_1 = \frac{\pi d_1^2}{4} \quad \text{and} \quad A_2 = \frac{\pi d_2^2}{4} \]

5. **Substituting Areas into the Ratio**:
Now substituting the expressions for \(A_1\) and \(A_2\) into the ratio:
\[ \frac{A_1^2}{A_2^2} = \frac{\left(\frac{\pi d_1^2}{4}\right)^2}{\left(\frac{\pi d_2^2}{4}\right)^2} = \frac{d_1^4}{d_2^4} \]
Therefore, we have:
\[ \frac{d_1^4}{d_2^4} = \frac{1}{4} \]

6. **Taking the Square Root**:
Taking the square root of both sides gives:
\[ \frac{d_1^2}{d_2^2} = \frac{1}{2} \]

7. **Finding the Diameter Ratio**:
Taking the square root again, we find:
\[ \frac{d_1}{d_2} = \frac{1}{\sqrt{2}} \]
Hence, the ratio of diameters \(d_1 : d_2\) is:
\[ d_1 : d_2 = \sqrt{2} : 1 \]

### Final Answer:
The ratio of the diameters of the two rods is \( \sqrt{2} : 1 \).