Two identical stringed instruments have frequency 100 Hz . If tension in one of them is increased by 4% and they are sounded together then the number of beats in one second is

Options

A. 1
B. 8
C. 4
D. 2

Answer

To solve the problem of finding the number of beats per second when the tension in one of the identical stringed instruments is increased by 4%, we can follow these steps:

### Step 1: Understand the Concept of Beats
Beats occur when two sound waves of slightly different frequencies interfere with each other. The number of beats per second is equal to the absolute difference in their frequencies.

### Step 2: Identify the Given Information
- Frequency of both instruments initially (f1 and f2) = 100 Hz
- Tension in one of the instruments is increased by 4%.

### Step 3: Calculate the New Frequency
The frequency of a string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Where:
- \( f \) = frequency
- \( T \) = tension
- \( \mu \) = linear mass density (constant for identical strings)
- \( L \) = length of the string (also constant for identical strings)

Since the length and mass density are constant, we can express the frequency in terms of tension:
\[ f \propto \sqrt{T} \]

### Step 4: Determine the New Tension
If the original tension is \( T \), then increasing it by 4% gives:
\[ T' = T + 0.04T = 1.04T \]

### Step 5: Calculate the New Frequency
Let the new frequency be \( f_2 \):
\[ f_2 = k \sqrt{T'} = k \sqrt{1.04T} = k \sqrt{1.04} \sqrt{T} \]
Here, \( k \) is a constant that incorporates \( \frac{1}{2L} \).

Since the original frequency \( f_1 = k \sqrt{T} = 100 \) Hz, we can relate the new frequency to the original:
\[ f_2 = 100 \sqrt{1.04} \]

### Step 6: Calculate \( \sqrt{1.04} \)
Using a calculator or approximation:
\[ \sqrt{1.04} \approx 1.02 \]

### Step 7: Find the New Frequency
Now substituting back:
\[ f_2 \approx 100 \times 1.02 = 102 \text{ Hz} \]

### Step 8: Calculate the Number of Beats
The number of beats per second is given by the absolute difference in frequencies:
\[ \text{Beats} = |f_2 - f_1| = |102 - 100| = 2 \text{ beats per second} \]

### Final Answer
The number of beats per second is **2 beats per second**.

---