The problem asks to calculate the frequency of the first harmonic for an air column closed at one end, given that the length of the air column is 60.0 cm and the temperature is 10.0°C.

Step 1: Formula for the First Harmonic of an Air Column Closed at One End

The fundamental frequency $f$ of an air column closed at one end is given by:

$f = \frac{v}{4L}$

Where:

• $f$ is the fundamental frequency (first harmonic).
• $v$ is the speed of sound in air.
• $L$ is the length of the air column.

Step 2: Speed of Sound in Air

The speed of sound in air depends on the temperature and is calculated by the following equation:

$v = 331.4 + 0.6T$

Where:

• $v$ is the speed of sound in m/s.
• $T$ is the temperature in °C.

For a temperature of $T = 10.0°C$:

$v = 331.4 + 0.6 \times 10 = 337.4 \, \text{m/s}$

Step 3: Calculation of the Frequency

Now, we know that:

• $v = 337.4 \, \text{m/s}$
• $L = 60.0 \, \text{cm} = 0.600 \, \text{m}$

Substituting these values into the formula for the first harmonic frequency:

$f = \frac{337.4}{4 \times 0.600}$

Let's calculate that next.The frequency of the first harmonic for the air column closed at one end is approximately 140.6 Hz.

Would you like further details or have any questions? Here are 5 related questions that can expand on this topic:

1. How does the speed of sound vary with temperature?
2. What would be the frequency if the air column were open at both ends?
3. How would the frequency change if the temperature increased?
4. What are higher harmonics and how do they differ from the fundamental frequency in air columns?
5. How is resonance achieved in air columns closed at one end?

Tip: For practical applications, the speed of sound in air increases with temperature because warmer air molecules move faster and transmit sound waves more efficiently.