For a vibrating tuning fork is put in contact with the sonometer box then the rider on the wire falls down. The frequency of tuning fork, as the compared to that of the sonometer wire will be -

Options

A. More
B. Less
C. Equal
D. No relation the two

Answer

To solve the problem, we need to analyze the situation involving a vibrating tuning fork and a sonometer wire. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Setup
We have a tuning fork that vibrates at a certain frequency (let's denote it as \( f \)). This tuning fork is placed in contact with a sonometer box, which contains a wire. The sonometer wire is capable of vibrating and producing sound waves.

**Hint:** Visualize the setup: a tuning fork producing sound and a wire that can vibrate when it resonates with that sound.

### Step 2: The Role of the Rider
The rider on the sonometer wire is a small weight that can be adjusted to change the tension in the wire. When the tuning fork is brought close to the sonometer, it causes the wire to vibrate.

**Hint:** Remember that the rider affects the tension in the wire, which in turn affects the frequency of the vibrations produced by the wire.

### Step 3: Resonance Condition
For the rider to fall off, the amplitude of the vibrations in the wire must be at its maximum. This occurs when there is resonance between the tuning fork and the sonometer wire. Resonance happens when the frequency of the tuning fork matches the natural frequency of the wire.

**Hint:** Think about what resonance means: it’s when two frequencies are equal, leading to maximum energy transfer.

### Step 4: Setting the Frequencies Equal
Let’s denote the frequency of the sonometer wire as \( f' \). For resonance to occur, the frequency of the tuning fork \( f \) must equal the frequency of the sonometer wire \( f' \). Therefore, we can write:

\[ f = f' \]

**Hint:** This step is crucial: if the frequencies are not equal, the rider will not fall, indicating that resonance is essential for the rider to drop.

### Step 5: Conclusion
Since the rider falls when the tuning fork is in contact with the sonometer box, we conclude that the frequency of the tuning fork is equal to the frequency of the sonometer wire.

Thus, the final answer is:

\[ \text{The frequency of the tuning fork is equal to that of the sonometer wire.} \]

### Summary
- The tuning fork vibrates at frequency \( f \).
- The sonometer wire vibrates at frequency \( f' \).
- For resonance, \( f = f' \).
- The rider falls when \( f = f' \), indicating resonance.