Using a tuning fork of frequency 512 Hz, the string of a sorcmeter has to be tuned. When they are vibrated together they produce 10 beats per second. By what percentage should the length of the string he altered to achieve tuning, the tension remaining the same.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given information
- The frequency of the tuning fork (n1) = 512 Hz
- The number of beats produced = 10 beats per second

### Step 2: Determine the frequency of the string (n2)
The frequency of the string (n2) can be calculated using the formula for beats:
\[ n2 = n1 + \text{number of beats} \]
Thus,
\[ n2 = 512 \, \text{Hz} + 10 \, \text{Hz} = 522 \, \text{Hz} \]

### Step 3: Use the relationship between frequency and length
The frequency of a vibrating string is inversely proportional to its length (L). Therefore, we can write:
\[ \frac{n1}{n2} = \frac{L2}{L1} \]
Where:
- L1 = original length of the string
- L2 = new length of the string after tuning

### Step 4: Substitute the known values
Substituting the values of n1 and n2:
\[ \frac{512}{522} = \frac{L2}{L1} \]

### Step 5: Rearrange to find L2
Cross-multiplying gives:
\[ L2 = L1 \cdot \frac{512}{522} \]

### Step 6: Calculate the percentage change in length
The percentage alteration in length can be calculated using the formula:
\[ \text{Percentage alteration} = \frac{L2 - L1}{L1} \times 100 \]
Substituting for L2:
\[ \text{Percentage alteration} = \frac{L1 \cdot \frac{512}{522} - L1}{L1} \times 100 \]
This simplifies to:
\[ \text{Percentage alteration} = \left(\frac{512}{522} - 1\right) \times 100 \]

### Step 7: Calculate the numerical value
Calculating the fraction:
\[ \frac{512}{522} - 1 = \frac{512 - 522}{522} = \frac{-10}{522} \]
Now, calculate the percentage:
\[ \text{Percentage alteration} = \left(\frac{-10}{522}\right) \times 100 \approx -1.92\% \]

### Conclusion
The length of the original string should be decreased by approximately **1.92%** to achieve tuning.

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