A cylindrical metal tube has a length of 50 cm and is open at both ends. Find the frequencies between 1000 Hz and 2000 Hz at which the air column in the tube can resonate. Speed of sound in air is `340 m s^-1`.

Explanation

To find the frequencies at which the air column in a cylindrical metal tube (open at both ends) can resonate, we can follow these steps:

### Step 1: Understand the Resonance in Open Tubes
In an open tube, the resonant frequencies can be determined using the formula:
\[ n \lambda = 2L \]
where:
- \( n \) is the harmonic number (1, 2, 3, ...),
- \( \lambda \) is the wavelength,
- \( L \) is the length of the tube.

### Step 2: Calculate the Wavelength
From the formula above, we can express the wavelength in terms of the length of the tube:
\[ \lambda = \frac{2L}{n} \]

### Step 3: Calculate the Frequency
The frequency \( f \) can be calculated using the speed of sound \( v \) and the wavelength \( \lambda \):
\[ f = \frac{v}{\lambda} \]
Substituting the expression for \( \lambda \):
\[ f = \frac{v}{\frac{2L}{n}} = \frac{vn}{2L} \]

### Step 4: Substitute Known Values
Given:
- Length \( L = 50 \) cm = 0.5 m,
- Speed of sound \( v = 340 \) m/s.

Substituting these values into the frequency formula:
\[ f = \frac{340 \cdot n}{2 \cdot 0.5} = 340n \]

### Step 5: Find Frequencies Between 1000 Hz and 2000 Hz
Now we need to find the values of \( n \) such that the frequency \( f \) lies between 1000 Hz and 2000 Hz:
\[ 1000 < 340n < 2000 \]

Dividing the entire inequality by 340:
\[ \frac{1000}{340} < n < \frac{2000}{340} \]
Calculating the bounds:
\[ 2.94 < n < 5.88 \]

Since \( n \) must be a whole number, the possible values for \( n \) are 3, 4, and 5.

### Step 6: Calculate the Frequencies
Now we can calculate the frequencies for \( n = 3, 4, 5 \):
- For \( n = 3 \):
\[ f = 340 \cdot 3 = 1020 \text{ Hz} \]

- For \( n = 4 \):
\[ f = 340 \cdot 4 = 1360 \text{ Hz} \]

- For \( n = 5 \):
\[ f = 340 \cdot 5 = 1700 \text{ Hz} \]

### Step 7: List the Frequencies
The frequencies that lie between 1000 Hz and 2000 Hz are:
- 1020 Hz
- 1360 Hz
- 1700 Hz

### Final Answer
The frequencies at which the air column in the tube can resonate between 1000 Hz and 2000 Hz are **1020 Hz, 1360 Hz, and 1700 Hz**.

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