Two waves of wavelength 50 cm and 51 cm produce 12 beat/s . The speed of sound is

Options

A. 1. 306 m/s
B. 2. 331 m/s
C. 3. 340 m/s
D. 4. 360 m/s

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
- Wavelength of the first wave, λ₁ = 50 cm = 0.50 m
- Wavelength of the second wave, λ₂ = 51 cm = 0.51 m
- Beat frequency, f_beat = 12 beats/s

### Step 2: Relate the frequencies of the waves to their wavelengths
The frequency (ν) of a wave is given by the formula:
\[ \nu = \frac{v}{\lambda} \]
where \( v \) is the speed of sound and \( \lambda \) is the wavelength.

For the two waves, we can write:
- Frequency of the first wave, \( \nu_1 = \frac{v}{\lambda_1} = \frac{v}{0.50} \)
- Frequency of the second wave, \( \nu_2 = \frac{v}{\lambda_2} = \frac{v}{0.51} \)

### Step 3: Set up the equation for beat frequency
The beat frequency is the absolute difference between the two frequencies:
\[ |\nu_1 - \nu_2| = f_{beat} \]
Substituting the expressions for \( \nu_1 \) and \( \nu_2 \):
\[ \left| \frac{v}{0.50} - \frac{v}{0.51} \right| = 12 \]

### Step 4: Simplify the equation
We can factor out \( v \):
\[ v \left( \frac{1}{0.50} - \frac{1}{0.51} \right) = 12 \]

### Step 5: Calculate the difference in frequencies
Calculating \( \frac{1}{0.50} - \frac{1}{0.51} \):
\[ \frac{1}{0.50} = 2 \quad \text{and} \quad \frac{1}{0.51} \approx 1.9608 \]
Thus,
\[ 2 - 1.9608 \approx 0.0392 \]

### Step 6: Substitute back to find speed of sound
Now we substitute this back into the equation:
\[ v \cdot 0.0392 = 12 \]
\[ v = \frac{12}{0.0392} \]

### Step 7: Calculate the speed
Calculating \( v \):
\[ v \approx 306.12 \, \text{m/s} \]
Rounding to two decimal places, we get:
\[ v \approx 306 \, \text{m/s} \]

### Final Answer
The speed of sound is approximately **306 m/s**.

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