Estimation of frequency of a wave forming a standing wave represented by `y = A sin kx cos t` can be done if the speed and wavelength are known using `speed = "Frequency" xx "wavelength"` . Speed of motion depends on the medium properties namely tension in string and mass per unit length of string . A string may vibrate with different frequencies . The corresponding wavelength should be related to the length of the string by a whole number for a string fixed at both ends . Answer the following questions:
A string fixed at both ends having a third overtone frequency of `200 Hz` while carrying a wave at a speed of `30 ms^(-1)` has a length of

Options

A. `30 m`
B. `22.5 cm`
C. `30 cm`
D. `10.25 cm`

Explanation

To find the length of a string fixed at both ends that has a third overtone frequency of 200 Hz and carries a wave at a speed of 30 m/s, we can follow these steps:

### Step 1: Understand the relationship between frequency, wavelength, and speed
The speed of a wave on a string is given by the formula:
\[ \text{Speed} = \text{Frequency} \times \text{Wavelength} \]
This can be rearranged to find the wavelength:
\[ \text{Wavelength} = \frac{\text{Speed}}{\text{Frequency}} \]

### Step 2: Identify the overtone
The third overtone corresponds to the fourth harmonic (n = 4) for a string fixed at both ends. The frequency of the nth harmonic is given by:
\[ f_n = \frac{n \cdot v}{2L} \]
Where:
- \( f_n \) is the frequency of the nth harmonic
- \( n \) is the harmonic number (4 for the third overtone)
- \( v \) is the speed of the wave
- \( L \) is the length of the string

### Step 3: Substitute the known values
We know:
- \( f_4 = 200 \, \text{Hz} \)
- \( v = 30 \, \text{m/s} \)
- \( n = 4 \)

Substituting these values into the harmonic frequency formula:
\[ 200 = \frac{4 \cdot 30}{2L} \]

### Step 4: Solve for L
Rearranging the equation to solve for \( L \):
\[ 200 = \frac{120}{2L} \]
\[ 200 = \frac{60}{L} \]
Multiplying both sides by \( L \):
\[ 200L = 60 \]
Now, divide both sides by 200:
\[ L = \frac{60}{200} \]
\[ L = 0.3 \, \text{m} \]

### Step 5: Convert to centimeters
Since the question asks for the length in centimeters:
\[ L = 0.3 \, \text{m} = 30 \, \text{cm} \]

### Final Answer
The length of the string is \( 30 \, \text{cm} \).

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