In a closed end pipe of length `105 cm` , standing waves are set up corresponding to the third overtone . What distance from the closed end, amongst the following is a pressure node?

Options

A. `(a)20 cm`
B. `(b)60 cm`
C. `(c)85 cm`
D. `(d)45 cm`

Explanation

To solve the problem of finding the distance from the closed end of a closed-end pipe where a pressure node occurs for the third overtone, we can follow these steps:

### Step 1: Understand the concept of standing waves in a closed-end pipe
In a closed-end pipe, standing waves are formed with one end closed (which is a displacement node) and the other end open (which is a displacement antinode). The pressure nodes occur at the displacement antinodes.

### Step 2: Identify the overtone and harmonics
The third overtone corresponds to the fourth harmonic. In a closed-end pipe, the harmonics are given by:
- 1st harmonic (fundamental frequency): 1/4 wavelength (λ/4)
- 2nd harmonic (1st overtone): 3/4 wavelength (3λ/4)
- 3rd harmonic (2nd overtone): 5/4 wavelength (5λ/4)
- 4th harmonic (3rd overtone): 7/4 wavelength (7λ/4)

### Step 3: Set up the equation for the length of the pipe
The length of the pipe (L) is given as 105 cm. For the third overtone (4th harmonic), the length of the pipe can be expressed as:
\[ L = \frac{7\lambda}{4} \]
Thus, we can write:
\[ \frac{7\lambda}{4} = 105 \]

### Step 4: Solve for the wavelength (λ)
To find the wavelength, rearrange the equation:
\[ 7\lambda = 105 \times 4 \]
\[ 7\lambda = 420 \]
\[ \lambda = \frac{420}{7} = 60 \text{ cm} \]

### Step 5: Determine the positions of pressure nodes
The pressure nodes occur at:
- 1st pressure node: \( \frac{\lambda}{4} = \frac{60}{4} = 15 \text{ cm} \)
- 2nd pressure node: \( \frac{3\lambda}{4} = \frac{3 \times 60}{4} = 45 \text{ cm} \)
- 3rd pressure node: \( \frac{5\lambda}{4} = \frac{5 \times 60}{4} = 75 \text{ cm} \)
- 4th pressure node: \( \frac{7\lambda}{4} = \frac{7 \times 60}{4} = 105 \text{ cm} \)

### Step 6: Identify the distance from the closed end
From the closed end, the distances to the pressure nodes are:
- 1st node: 15 cm
- 2nd node: 45 cm
- 3rd node: 75 cm
- 4th node: 105 cm

Among the options provided, the distance from the closed end that corresponds to a pressure node is **45 cm**.

### Final Answer
The distance from the closed end where there is a pressure node is **45 cm**.

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