When a certain string is clamped at both ends, the lowest four resonant frequencies are measured to be 100, 150, 200, and 250 Hz. One of the resonant frequencies (below 200 Hz) is missing. What is it?

Options

A. 25Hz
B. 50Hz
C. 75Hz
D. 125Hz

Answer

To find the missing resonant frequency below 200 Hz for the clamped string, we can follow these steps:

### Step 1: Understand the Harmonic Frequencies
When a string is clamped at both ends, it vibrates in harmonics. The frequencies of the harmonics can be expressed as:
- Fundamental frequency (1st harmonic): \( F_1 \)
- 2nd harmonic: \( F_2 = 2F_1 \)
- 3rd harmonic: \( F_3 = 3F_1 \)
- 4th harmonic: \( F_4 = 4F_1 \)
- and so on.

### Step 2: Identify Given Frequencies
The measured resonant frequencies are:
- 100 Hz
- 150 Hz
- 200 Hz
- 250 Hz

### Step 3: Analyze the Frequencies
From the given frequencies, we can see that:
- 100 Hz could be \( F_2 \) (2nd harmonic)
- 150 Hz could be \( F_3 \) (3rd harmonic)
- 200 Hz could be \( F_4 \) (4th harmonic)
- 250 Hz could be \( F_5 \) (5th harmonic)

### Step 4: Determine the Fundamental Frequency
To find the fundamental frequency \( F_1 \), we can use the relationships between the harmonics:
- If \( F_2 = 100 \) Hz, then \( F_1 = \frac{F_2}{2} = \frac{100}{2} = 50 \) Hz.
- If \( F_3 = 150 \) Hz, then \( F_1 = \frac{F_3}{3} = \frac{150}{3} = 50 \) Hz.
- If \( F_4 = 200 \) Hz, then \( F_1 = \frac{F_4}{4} = \frac{200}{4} = 50 \) Hz.
- If \( F_5 = 250 \) Hz, then \( F_1 = \frac{F_5}{5} = \frac{250}{5} = 50 \) Hz.

In all cases, we find that \( F_1 = 50 \) Hz.

### Step 5: Identify the Missing Frequency
Now we can find the missing frequency below 200 Hz:
- The harmonics would be:
- \( F_1 = 50 \) Hz (1st harmonic)
- \( F_2 = 100 \) Hz (2nd harmonic)
- \( F_3 = 150 \) Hz (3rd harmonic)
- \( F_4 = 200 \) Hz (4th harmonic)
- \( F_5 = 250 \) Hz (5th harmonic)

The missing frequency below 200 Hz is \( F_1 = 50 \) Hz.

### Conclusion
The missing resonant frequency below 200 Hz is **50 Hz**.

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