When a mass m is hung on a spring, the spring stretched by 6 cm. If the loaded spring is pulled downward a little and released, then the period of vibration of the system will be

Options

A. 0.27 s
B. 0.35 s
C. 0.49 s
D. 0.64 s

Explanation

To solve the problem of finding the period of vibration of a mass-spring system when a mass \( m \) is hung on a spring that stretches by 6 cm, we can follow these steps:

### Step-by-Step Solution

1. **Convert the Stretch to Meters**:
The spring stretches by 6 cm. We need to convert this to meters for our calculations.
\[ \text{Stretch} = 6 \, \text{cm} = 6 \times 10^{-2} \, \text{m} \]

2. **Identify the Formula for the Period of Vibration**:
The period \( T \) of a mass-spring system undergoing simple harmonic motion is given by the formula:
\[ T = 2\pi \sqrt{\frac{x}{g}} \]
where \( x \) is the stretch of the spring (in meters) and \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

3. **Substitute the Values into the Formula**:
Now, we can substitute the values we have into the formula:
\[ T = 2\pi \sqrt{\frac{6 \times 10^{-2}}{10}} \]

4. **Simplify the Expression**:
First, simplify the fraction inside the square root:
\[ \frac{6 \times 10^{-2}}{10} = 6 \times 10^{-3} \]
Now substitute this back into the equation:
\[ T = 2\pi \sqrt{6 \times 10^{-3}} \]

5. **Calculate the Square Root**:
Calculate the square root:
\[ \sqrt{6 \times 10^{-3}} = \sqrt{6} \times \sqrt{10^{-3}} = \sqrt{6} \times 10^{-1.5} \]
Approximating \( \sqrt{6} \approx 2.45 \):
\[ \sqrt{6 \times 10^{-3}} \approx 2.45 \times 0.03162 \approx 0.0775 \]

6. **Calculate the Period**:
Now, substitute this back into the equation for \( T \):
\[ T \approx 2\pi \times 0.0775 \approx 0.486 \, \text{seconds} \]
Rounding this value gives us approximately:
\[ T \approx 0.49 \, \text{seconds} \]

7. **Final Answer**:
Thus, the period of vibration of the system is approximately \( 0.49 \, \text{seconds} \).

### Conclusion
The correct option for the period of vibration is **0.49 seconds**.