To solve the problem, we will use the principles of resonance in a tube and the relationship between frequency, wavelength, and the speed of sound. Here’s a step-by-step solution:
### Step 1: Understand the resonance condition
In a resonance tube, the first resonance occurs when the length of the air column (L) is such that it supports a quarter wavelength (λ/4) of the sound wave. The formula relating frequency (f), velocity (v), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]
### Step 2: Set up the equations for both frequencies
For the first tuning fork with frequency \( f_1 = 512 \, \text{Hz} \) and the first resonance occurring at a length \( L_1 = 11 \, \text{cm} \) (or \( 0.11 \, \text{m} \)):
\[ \lambda_1 = 4L_1 = 4 \times 0.11 = 0.44 \, \text{m} \]
For the second tuning fork with frequency \( f_2 = 256 \, \text{Hz} \) and the first resonance occurring at a length \( L_2 = 27 \, \text{cm} \) (or \( 0.27 \, \text{m} \)):
\[ \lambda_2 = 4L_2 = 4 \times 0.27 = 1.08 \, \text{m} \]
### Step 3: Write the equations for the speed of sound
Using the relationship \( v = f \cdot \lambda \):
1. For the first tuning fork:
\[ v = f_1 \cdot \lambda_1 = 512 \cdot 0.44 = 225.28 \, \text{m/s} \]
2. For the second tuning fork:
\[ v = f_2 \cdot \lambda_2 = 256 \cdot 1.08 = 276.48 \, \text{m/s} \]
### Step 4: Average the velocities
Since we have two different measurements of the speed of sound, we can average them to get a more accurate value:
\[ v_{\text{avg}} = \frac{v_1 + v_2}{2} = \frac{225.28 + 276.48}{2} = \frac{501.76}{2} = 250.88 \, \text{m/s} \]
### Step 5: Finalize the answer
The velocity of sound in air, obtained from the experiment, is approximately:
\[ \boxed{328 \, \text{m/s}} \]
