To understand how the vibration of a sonometer wire changes when its length is doubled, we need to delve into the relationship between the length of the wire, its tension, and the frequency of the sound it produces. The fundamental frequency of a vibrating string or wire is determined by its length, tension, and mass per unit length. When the length of the wire is increased while keeping the tension constant, the frequency decreases. This is because the fundamental frequency is inversely proportional to the length of the wire.
The Fundamental Mode of Vibration
In the fundamental mode, the wire vibrates in a single segment, creating one antinode in the center and two nodes at the ends. The frequency of this mode can be expressed as:
Where:
- f = frequency
- L = length of the wire
- T = tension in the wire
- μ = mass per unit length of the wire
Doubling the Length
When the length of the wire is doubled (let's say from L to 2L), the new fundamental frequency becomes:
- f' = (1/2(2L)) * √(T/μ) = (1/4L) * √(T/μ)
This shows that the new fundamental frequency is half of the original frequency. To maintain resonance with the same tuning fork, which produces a specific frequency, the wire must vibrate in a mode that matches this frequency.
Higher Harmonics
When the length of the wire is increased, it can still vibrate in higher harmonic modes. The harmonics of a string are integer multiples of the fundamental frequency. The second harmonic (first overtone) divides the wire into two segments, the third harmonic into three segments, and so on.
Finding the Correct Mode
Since the fundamental frequency has decreased, we need to find a mode where the frequency matches that of the tuning fork. The relationship between the mode number (n) and the frequency can be expressed as:
- f_n = n * (1/2L) * √(T/μ)
For the new length (2L), the frequency for the nth harmonic becomes:
- f_n' = n * (1/4L) * √(T/μ)
To find the mode that can resonate with the original frequency, we can set:
From this, we can see that if the original fundamental frequency has become half, the wire can now vibrate in the second harmonic (2 segments) to match the original frequency. However, it can also vibrate in higher harmonics.
Conclusion
Given that the wire can vibrate in higher harmonics, the correct answer to your question is that the wire can vibrate in 2 segments (the second harmonic) to remain in resonance with the tuning fork. Therefore, the answer is:
