A tuning fork of frequency 480 Hz produces 10 beats per second when sounded with a vibrating sonometer wire. What must have been the frequency of string if a slight increase in tension produces fewer beats second than before.

Options

A. 460 Hz
B. 470 Hz
C. 480 Hz
D. 490 Hz

Explanation

To solve the problem step by step, we will analyze the information given and apply the concepts of beats and frequency.

### Step 1: Understand the given data
- Frequency of the tuning fork (f_tuning) = 480 Hz
- Beat frequency (f_beat) = 10 Hz

### Step 2: Set up the equation for beat frequency
The beat frequency is given by the absolute difference between the frequencies of the two sources:
\[ f_{beat} = |f_{string} - f_{tuning}| \]
Let the frequency of the string be \( f_{string} \).

### Step 3: Set up the equations based on the beat frequency
From the beat frequency formula, we can write:
\[ |f_{string} - 480| = 10 \]
This gives us two possible equations:
1. \( f_{string} - 480 = 10 \)
2. \( f_{string} - 480 = -10 \)

### Step 4: Solve the equations
1. From the first equation:
\[ f_{string} = 480 + 10 = 490 \text{ Hz} \]
2. From the second equation:
\[ f_{string} = 480 - 10 = 470 \text{ Hz} \]

Thus, the possible frequencies for the string are 490 Hz and 470 Hz.

### Step 5: Analyze the effect of increasing tension
The problem states that a slight increase in tension produces fewer beats per second than before. This implies that the frequency of the string must decrease when tension is increased.

### Step 6: Determine which frequency is correct
- If \( f_{string} = 490 \) Hz and we increase the tension, the frequency of the string would increase (since tension increases the speed of the wave in the string), leading to a larger beat frequency.
- If \( f_{string} = 470 \) Hz and we increase the tension, the frequency of the string would also increase, but since it is already lower than the tuning fork frequency, the beat frequency would decrease.

### Conclusion
Since we need the frequency of the string that results in fewer beats per second when tension is increased, the correct frequency of the string is:
\[ \boxed{470 \text{ Hz}} \]