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 need to find the highest overtone of the sound produced by the steel piano wire that can be heard by a person who can hear frequencies up to 10 kHz.

### Step 1: Identify the given data
- Length of the wire (L) = 50 cm = 0.5 m
- Mass of the wire (m) = 5 g = 0.005 kg
- Tension (T) = 400 N
- Maximum frequency (f_max) = 10 kHz = 10,000 Hz

### Step 2: Calculate the linear mass density (μ) of the wire
The linear mass density (μ) is given by the formula:
\[ \mu = \frac{m}{L} \]
Substituting the values:
\[ \mu = \frac{0.005 \text{ kg}}{0.5 \text{ m}} = 0.01 \text{ kg/m} \]

### Step 3: Calculate the speed of the wave (v) in the wire
The speed of the wave 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: Determine the frequency of the overtones
The frequency of the nth overtone (f_n) is given by:
\[ f_n = \frac{n \cdot v}{2L} \]
Where n is the harmonic number (1 for the fundamental frequency, 2 for the first overtone, etc.).

### Step 5: Set up the equation for the maximum frequency
We need to find the highest overtone (n) that can be heard, so we set:
\[ f_n \leq f_{\text{max}} \]
Substituting the expression for f_n:
\[ \frac{n \cdot 200}{2 \cdot 0.5} \leq 10000 \]
This simplifies to:
\[ n \cdot 200 \leq 10000 \]
\[ n \leq \frac{10000}{200} = 50 \]

### Step 6: Determine the highest overtone
The highest overtone corresponds to n = 50. However, the overtone number is defined as n - 1, where n is the harmonic number. Therefore, the highest overtone is:
\[ \text{Highest overtone} = n - 1 = 50 - 1 = 49 \]

### 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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