A closed pipe and an open pipe sounding together produce 5 beats per second. If the length of the open pipe is 30 cm, find by how much the length of the closed pipe should be changed to bring the two pipes in unison. Take speed of sound in air is 330 m/s.

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have a closed pipe and an open pipe producing 5 beats per second. The length of the open pipe (L0) is given as 30 cm, and we need to find how much the length of the closed pipe (Lc) should be changed to bring both pipes into unison.

### Step 2: Convert Units
Convert the length of the open pipe from centimeters to meters for consistency with the speed of sound.
- L0 = 30 cm = 0.30 m

### Step 3: Calculate Frequencies
The frequency of the closed pipe (fc) and the open pipe (fo) can be calculated using the formulas:
- For a closed pipe: \( f_c = \frac{v}{4L_c} \)
- For an open pipe: \( f_o = \frac{v}{2L_0} \)

Where \( v \) is the speed of sound in air, given as 330 m/s.

### Step 4: Set Up the Beat Frequency Equation
The beat frequency is the difference between the two frequencies:
\[ |f_o - f_c| = 5 \text{ beats/second} \]
This can be written as:
\[ \frac{v}{2L_0} - \frac{v}{4L_c} = 5 \]

### Step 5: Substitute Known Values
Substituting \( v = 330 \) m/s and \( L_0 = 0.30 \) m into the equation:
\[ \frac{330}{2 \times 0.30} - \frac{330}{4L_c} = 5 \]

### Step 6: Simplify the Equation
Calculate \( \frac{330}{2 \times 0.30} \):
\[ \frac{330}{0.60} = 550 \]
Thus, the equation becomes:
\[ 550 - \frac{330}{4L_c} = 5 \]

### Step 7: Solve for \( L_c \)
Rearranging gives:
\[ \frac{330}{4L_c} = 550 - 5 \]
\[ \frac{330}{4L_c} = 545 \]
Now, multiply both sides by \( 4L_c \):
\[ 330 = 545 \times 4L_c \]
\[ L_c = \frac{330}{2180} \]
Calculating this gives:
\[ L_c = 0.1514 \text{ m} \]

### Step 8: Find the New Length for Unison
When the pipes are in unison, the frequency will be the same:
\[ \frac{v}{2L_0} = \frac{v}{4L_c'} \]
This implies:
\[ 2L_0 = 4L_c' \]
\[ L_c' = \frac{L_0}{2} = \frac{0.30}{2} = 0.15 \text{ m} \]

### Step 9: Calculate the Change in Length
Now, we find the change in length of the closed pipe:
\[ \Delta L_c = L_c - L_c' \]
Substituting the values:
\[ \Delta L_c = 0.1514 - 0.15 = 0.0014 \text{ m} \]

### Final Answer
The length of the closed pipe should be changed by approximately 0.0014 m (or 1.4 mm) to bring the two pipes in unison.

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