An open pipe resonates with a tuning fork of frequency `500 Hz` . It is observed that two successive notes are formed at distance `16 and 46 cm` from the open end. The speed of sound in air in the pipe is

Options

A. `230 m//s`
B. `300 m//s`
C. `320 m//s`
D. `360 m//s`

Explanation

To find the speed of sound in air in the pipe, we can follow these steps:

### Step 1: Understand the problem
We have an open pipe that resonates with a tuning fork of frequency \( f = 500 \, \text{Hz} \). Two successive nodes are observed at distances \( d_1 = 16 \, \text{cm} \) and \( d_2 = 46 \, \text{cm} \) from the open end.

### Step 2: Identify the positions of nodes
In an open pipe, the positions of the nodes can be expressed in terms of the wavelength \( \lambda \):
- The first node (at \( d_1 \)) corresponds to \( \frac{\lambda}{4} \).
- The second node (at \( d_2 \)) corresponds to \( \frac{3\lambda}{4} \).

### Step 3: Set up the equations
From the distances given:
1. For the first node:
\[ \frac{\lambda}{4} = 16 \, \text{cm} \]
Therefore, we can express \( \lambda \):
\[ \lambda = 4 \times 16 \, \text{cm} = 64 \, \text{cm} \]

2. For the second node:
\[ \frac{3\lambda}{4} = 46 \, \text{cm} \]
Thus, we can also express \( \lambda \):
\[ \lambda = \frac{4 \times 46}{3} \, \text{cm} = \frac{184}{3} \, \text{cm} \approx 61.33 \, \text{cm} \]

### Step 4: Calculate the difference in distance
The distance between the two nodes is:
\[ d_2 - d_1 = 46 \, \text{cm} - 16 \, \text{cm} = 30 \, \text{cm} \]
This distance corresponds to \( \frac{\lambda}{2} \) (since it represents one full wavelength from one node to the next).

### Step 5: Find the wavelength
From the difference in distance:
\[ \frac{\lambda}{2} = 30 \, \text{cm} \implies \lambda = 60 \, \text{cm} = 0.6 \, \text{m} \]

### Step 6: Calculate the speed of sound
The speed of sound in the pipe can be calculated using the formula:
\[ v = f \cdot \lambda \]
Substituting the values we have:
\[ v = 500 \, \text{Hz} \times 0.6 \, \text{m} = 300 \, \text{m/s} \]

### Conclusion
The speed of sound in air in the pipe is \( 300 \, \text{m/s} \).

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