A smoked plate falls vertically under gravity. A tuning fork traces wave on it. It is found that the lengths of two conscutive grounps of 10 waves are 5.143 and 6.64 cm respectively. What is the frequecny of the fork ?

Answer

To find the frequency of the tuning fork based on the given lengths of two consecutive groups of 10 waves, we can follow these steps:

### Step 1: Identify the lengths of the wave groups
The problem states that the lengths of two consecutive groups of 10 waves are:
- Length of first group (L1) = 5.143 cm
- Length of second group (L2) = 6.64 cm

### Step 2: Calculate the wavelength (λ)
The distance between the two groups of waves represents the wavelength (λ) of the sound produced by the tuning fork. To find λ, we subtract the length of the first group from the length of the second group:
\[ \lambda = L2 - L1 = 6.64 \, \text{cm} - 5.143 \, \text{cm} = 1.497 \, \text{cm} \]

### Step 3: Convert the wavelength to meters
Since the standard unit for wavelength in physics is meters, we need to convert the wavelength from centimeters to meters:
\[ \lambda = 1.497 \, \text{cm} = 1.497 \times 10^{-2} \, \text{m} \]

### Step 4: Use the speed of sound to find the frequency (f)
The frequency (f) can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
where \( v \) is the speed of sound in air, approximately \( 340 \, \text{m/s} \).

Substituting the values:
\[ f = \frac{340 \, \text{m/s}}{1.497 \times 10^{-2} \, \text{m}} \approx 22700.67 \, \text{Hz} \]

### Step 5: Round the frequency
Rounding the frequency to two decimal places, we get:
\[ f \approx 22700 \, \text{Hz} \]

### Final Answer
The frequency of the tuning fork is approximately **22700 Hz**.

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