Rituraj Tiwari
Last Activity: 5 Years ago
To understand how a sonometer wire can maintain resonance with a tuning fork when the length is doubled, let's break down the concepts involved in this situation. The sonometer is a device used to study the relationship between tension, length, and frequency of a vibrating wire, and it operates based on the principles of standing waves.
The Basics of Vibrating Wires
When a wire is fixed at both ends, it can vibrate to produce sound waves. The fundamental frequency, or first harmonic, occurs when the wire vibrates in a single segment, creating one antinode in the middle and two nodes at the ends. The frequency of vibration depends on three key factors:
- Tension (T): The force applied along the length of the wire.
- Length (L): The distance between the fixed points of the wire.
- Linear density (μ): The mass per unit length of the wire.
Understanding the Relationship
The fundamental frequency (f) of the vibrating wire can be expressed by the formula:
f = (1/2L) * √(T/μ)
From this equation, we see that frequency is inversely proportional to the length of the wire. Therefore, if the length of the wire is doubled while keeping tension and linear density constant, the fundamental frequency will be halved. This means that the original tuning fork will no longer resonate with the wire vibrating in the fundamental mode.
Introducing Higher Modes of Vibration
Now, if we double the length of the wire, it can no longer vibrate in its fundamental mode to resonate with the tuning fork. However, we can achieve resonance by changing the mode of vibration. When the wire vibrates in a higher harmonic, it can accommodate the longer length while still producing sound waves that can match the frequency of the tuning fork.
Segmentation of the Wire
When the wire is vibrating in 6 segments, it is actually vibrating in the third harmonic mode. In this mode, the wire has 3 antinodes and 4 nodes. The relationship for the frequency of the nth harmonic can be expressed as:
f_n = (n/2L) * √(T/μ)
For the third harmonic (n = 3), the frequency becomes:
f_3 = (3/2(2L)) * √(T/μ) = (3/4L) * √(T/μ)
Here, we see that the frequency for the third harmonic is still dependent on the length, but since we have effectively doubled the length of the wire, we can see that the tuning fork can still match the frequency of the wire when it vibrates in this mode.
Conclusion
In summary, when the length of the sonometer wire is doubled, it can no longer vibrate in the fundamental mode to maintain resonance with a tuning fork. However, by vibrating in a higher harmonic (in this case, the third harmonic with 6 segments), the wire can still achieve resonance with the tuning fork. This demonstrates the fascinating interplay between length, tension, and frequency in vibrating systems.
