The young modulii of two springs of equal length and equal cross sectional areas are in the ratio 2:3 both the springs are suspended and loaded with the same mass . When springs are stretched and released , the time period of oscillation of the two springs is in the ratio of

Options

A. 3:2
B. 2:3
C. `sqrt3:sqrt2`
D. `sqrt2:sqrt3`

Explanation

To solve the problem of finding the ratio of the time periods of oscillation of two springs with given Young's moduli, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between Young's modulus and spring constant (k)**:
The spring constant \( k \) is related to Young's modulus \( Y \) by the formula:
\[ k = \frac{Y \cdot A}{L} \]
where \( A \) is the cross-sectional area and \( L \) is the length of the spring. Since both springs have equal lengths and equal cross-sectional areas, we can express the spring constants in terms of Young's moduli:
\[ k_1 = \frac{Y_1 \cdot A}{L} \quad \text{and} \quad k_2 = \frac{Y_2 \cdot A}{L} \]

2. **Calculate the ratio of spring constants**:
From the above equations, we can find the ratio of the spring constants:
\[ \frac{k_1}{k_2} = \frac{Y_1}{Y_2} \]

3. **Use the time period formula for oscillations**:
The time period \( T \) of oscillation for a mass \( m \) attached to a spring is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Therefore, the time periods for the two springs can be expressed as:
\[ T_1 = 2\pi \sqrt{\frac{m}{k_1}} \quad \text{and} \quad T_2 = 2\pi \sqrt{\frac{m}{k_2}} \]

4. **Find the ratio of the time periods**:
To find the ratio of the time periods \( \frac{T_1}{T_2} \):
\[ \frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{m}{k_1}}}{2\pi \sqrt{\frac{m}{k_2}}} = \sqrt{\frac{k_2}{k_1}} \]

5. **Substitute the ratio of spring constants**:
Since we have \( \frac{k_1}{k_2} = \frac{Y_1}{Y_2} \), we can write:
\[ \frac{T_1}{T_2} = \sqrt{\frac{Y_2}{Y_1}} \]

6. **Use the given ratio of Young's moduli**:
The problem states that the Young's moduli are in the ratio \( \frac{Y_1}{Y_2} = \frac{2}{3} \). Thus:
\[ \frac{Y_2}{Y_1} = \frac{3}{2} \]

7. **Final calculation of the time period ratio**:
Substituting this into our equation gives:
\[ \frac{T_1}{T_2} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} \]

### Final Answer:
The ratio of the time periods of oscillation of the two springs is:
\[ \frac{T_1}{T_2} = \frac{\sqrt{3}}{\sqrt{2}} \]