The vibrational frequency of a string is 300 Hz. The tension in the string is made twice and the length of the string is changed in such a way that it vibrates with a frequency of 600 Hz. What will be the ratio of new length to original length ?

Answer

To solve the problem, we need to analyze the relationship between frequency, tension, and length of the string using the formula for the frequency of a vibrating string.

### Step-by-Step Solution:

1. **Identify the initial conditions:**
- The initial frequency \( f_1 = 300 \, \text{Hz} \)
- Let the initial tension be \( T \)
- Let the initial length be \( L \)
- The mass per unit length is denoted as \( \nu \).

2. **Use the frequency formula for the initial conditions:**
The frequency of a vibrating string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\nu}} \]
For the initial conditions:
\[ 300 = \frac{1}{2L} \sqrt{\frac{T}{\nu}} \tag{1} \]

3. **Identify the new conditions:**
- The new frequency \( f_2 = 600 \, \text{Hz} \)
- The tension is doubled, so the new tension \( T' = 2T \)
- Let the new length be \( L' \).

4. **Use the frequency formula for the new conditions:**
For the new conditions:
\[ 600 = \frac{1}{2L'} \sqrt{\frac{2T}{\nu}} \tag{2} \]

5. **Rearranging equation (1):**
From equation (1):
\[ 300 \cdot 2L = \sqrt{\frac{T}{\nu}} \implies 600L = \sqrt{\frac{T}{\nu}} \tag{3} \]

6. **Rearranging equation (2):**
From equation (2):
\[ 600 \cdot 2L' = \sqrt{\frac{2T}{\nu}} \implies 1200L' = \sqrt{\frac{2T}{\nu}} \tag{4} \]

7. **Equating the two expressions:**
Now, we can square both sides of equations (3) and (4) to eliminate the square root:
- From (3):
\[ (600L)^2 = \frac{T}{\nu} \implies 360000L^2 = \frac{T}{\nu} \]
- From (4):
\[ (1200L')^2 = \frac{2T}{\nu} \implies 1440000(L')^2 = \frac{2T}{\nu} \]

8. **Setting the equations equal:**
Now we can set the two expressions for \( \frac{T}{\nu} \) equal to each other:
\[ 360000L^2 = \frac{1}{2} \cdot 1440000(L')^2 \]
Simplifying gives:
\[ 360000L^2 = 720000(L')^2 \]

9. **Finding the ratio of lengths:**
Dividing both sides by \( 720000 \):
\[ \frac{L^2}{(L')^2} = \frac{360000}{720000} = \frac{1}{2} \]
Taking the square root of both sides:
\[ \frac{L}{L'} = \frac{1}{\sqrt{2}} \implies \frac{L'}{L} = \sqrt{2} \]

10. **Final ratio:**
Therefore, the ratio of the new length to the original length is:
\[ \frac{L'}{L} = \sqrt{2} \approx 1.414 \]

### Conclusion:
The ratio of the new length to the original length is \( \sqrt{2} : 1 \).