Question

use the guess method.
14. A tuning fork with a frequency of 1024 Hz is held at the mouth of a cylinder partially filled with water. If the air temperature is 30°C, what is the minimum length of air that would resonate?

Answer 1

To find the minimum length of air that resonates with a tuning fork of frequency \( f = 1024 \) Hz, we can use the formula for the speed of sound in air and the properties of standing waves in a tube.

### Step 1: Calculate the Speed of Sound in Air

The speed of sound in air can be estimated using the formula:

\[
v = 331.4 + 0.6 \cdot T
\]

where \( T \) is the temperature in degrees Celsius.

For \( T = 30^\circ \text{C} \):

\[
v = 331.4 + 0.6 \cdot 30 = 331.4 + 18 = 349.4 \, \text{m/s}
\]

### Step 2: Determine the Wavelength

The speed of sound \( v \) can also be related to frequency \( f \) and wavelength \( \lambda \) using the formula:

\[
v = f \cdot \lambda
\]

Rearranging this gives us:

\[
\lambda = \frac{v}{f}
\]

Substituting the values:

\[
\lambda = \frac{349.4 \, \text{m/s}}{1024 \, \text{Hz}} \approx 0.341 \, \text{m}
\]

### Step 3: Find the Minimum Length of Resonating Air Column

In a tube closed at one end (like our partially filled cylinder), the minimum length of the air column that resonates corresponds to the fundamental frequency, which is a quarter of the wavelength:

\[
L = \frac{\lambda}{4}
\]

Now substituting the wavelength we calculated:

\[
L = \frac{0.341 \, \text{m}}{4} \approx 0.08525 \, \text{m}
\]

### Conclusion

Thus, the minimum length of the air column that would resonate with the tuning fork is approximately:

\[
\boxed{0.085 \, \text{m}} \quad \text{or} \quad \approx 8.5 \, \text{cm}
\]