A person can hear frequencies only upto 10kHz. A steel piano pipe wire 50cm long of mass 5g is streched with a tension of 400N. The number of the hightest overtone of the sound produced by this plano wire that the person can hear is

Options

A. 48
B. 50
C. 49
D. 51

Answer

To solve the problem step by step, we will follow the principles of wave motion and the properties of vibrating strings.

### Step 1: Understand the given data
- Maximum frequency a person can hear, \( f_{max} = 10 \, \text{kHz} = 10 \times 10^3 \, \text{Hz} \)
- Length of the wire, \( L = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Mass of the wire, \( m = 5 \, \text{g} = 5 \times 10^{-3} \, \text{kg} \)
- Tension in the wire, \( T = 400 \, \text{N} \)

### Step 2: Calculate the linear mass density (\( \mu \))
The linear mass density \( \mu \) is given by:
\[ \mu = \frac{m}{L} \]
Substituting the values:
\[ \mu = \frac{5 \times 10^{-3} \, \text{kg}}{0.5 \, \text{m}} = 0.01 \, \text{kg/m} \]

### Step 3: Calculate the wave speed (\( V \))
The wave speed \( V \) in the wire can be calculated using the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]
Substituting the values:
\[ V = \sqrt{\frac{400 \, \text{N}}{0.01 \, \text{kg/m}}} = \sqrt{40000} = 200 \, \text{m/s} \]

### Step 4: Calculate the fundamental frequency (\( f_1 \))
The fundamental frequency \( f_1 \) is given by the formula:
\[ f_1 = \frac{V}{2L} \]
Substituting the values:
\[ f_1 = \frac{200 \, \text{m/s}}{2 \times 0.5 \, \text{m}} = \frac{200}{1} = 200 \, \text{Hz} \]

### Step 5: Calculate the overtone frequencies
The frequency of the \( n \)-th overtone is given by:
\[ f_n = n \cdot f_1 \]
Thus, the maximum frequency that can be heard is:
\[ f_n \leq f_{max} \]
Substituting the values:
\[ n \cdot 200 \, \text{Hz} \leq 10 \times 10^3 \, \text{Hz} \]
\[ n \leq \frac{10 \times 10^3}{200} = 50 \]

### Step 6: Determine the highest overtone
The highest overtone is given by:
\[ \text{Highest overtone} = n - 1 \]
Substituting the value of \( n \):
\[ \text{Highest overtone} = 50 - 1 = 49 \]

### Final Answer
The number of the highest overtone of the sound produced by this piano wire that the person can hear is **49**.

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