A tuning fork of frequency n is held near the open end of tube, the tube is adjusted until resonance occurs. If the two shortest lengths to produce resonance are `L_1 and L_2` , then the speed of the sound is

Options

A. `n(L_2-L_1)`
B. `(n(L_2-L_1))/2`
C. `4n(L_2-L_1)`
D. `2n(L_2-L_1)`

Answer

To solve the problem, we need to find the speed of sound in a tube when resonance occurs at two different lengths, \( L_1 \) and \( L_2 \). Here’s a step-by-step solution:

### Step 1: Understanding the Resonance in the Tube
When a tuning fork of frequency \( n \) is held near the open end of a tube, resonance occurs at specific lengths of the tube. For a tube that is open at one end and closed at the other, the lengths at which resonance occurs can be described by the formula for the wavelengths of the sound.

### Step 2: Wavelength and Length Relationship
For a tube that is open at one end and closed at the other, the relationship between the length of the tube \( L \) and the wavelength \( \lambda \) of the sound wave is given by:
\[ L = \frac{(2n - 1) \lambda}{4} \]
where \( n \) is a positive integer (1, 2, 3, ...).

### Step 3: Expressing Wavelength in Terms of Lengths
For the two shortest lengths \( L_1 \) and \( L_2 \) that produce resonance, we can write:
- For \( L_1 \) (when \( n = 1 \)):
\[ L_1 = \frac{(2 \cdot 1 - 1) \lambda}{4} = \frac{\lambda}{4} \]
- For \( L_2 \) (when \( n = 2 \)):
\[ L_2 = \frac{(2 \cdot 2 - 1) \lambda}{4} = \frac{3\lambda}{4} \]

### Step 4: Finding the Difference in Lengths
Now, we find the difference between the two lengths:
\[ L_2 - L_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

### Step 5: Relating Wavelength to Speed of Sound
The speed of sound \( v \) is related to the wavelength \( \lambda \) and frequency \( n \) by the equation:
\[ v = n \lambda \]

### Step 6: Substituting Wavelength in Terms of Lengths
From the previous steps, we have:
\[ \lambda = 2(L_2 - L_1) \]
Substituting this into the speed of sound equation:
\[ v = n \cdot 2(L_2 - L_1) \]

### Step 7: Final Expression for Speed of Sound
Thus, we can express the speed of sound as:
\[ v = 2n(L_2 - L_1) \]

### Conclusion
The speed of sound in the tube is given by:
\[ v = 2n(L_2 - L_1) \]