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Consider the following statements-

In a stationary wave

1. All the particles perform simple harmonic motion with a frequency which is four times that of the two component waves.

2. Particles on the opposite sides of a node vibrate with a phase difference of π.

3. The amplitude of vibration of a particle at an antinode is equal to that of either component wave.

4. All the particles between two adjacent nodes vibrate in phase.

Of these statements-

  • a)
    2 and 4 are correct

  • b)
    1,3 and 4 are correct

  • c)
    1 and 2 are correct

  • d)
    1, 2, 3 and 4 are correct

Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Consider the following statements-In a stationary wave1. All the parti...
  1. All the particles perform simple harmonic motion with a frequency which is four times that of the two component waves: This statement is incorrect. In stationary waves, the frequency of the particle motion matches the frequency of the component waves that form the stationary wave. The frequency does not multiply.
  2. Particles on the opposite sides of a node vibrate with a phase difference of π: This statement is correct. Particles on opposite sides of a node in a stationary wave are out of phase by π (180 degrees). This means when one side is at its maximum positive displacement, the other side is at its maximum negative displacement.
  3. The amplitude of vibration of a particle at an antinode is equal to that of either component wave: This statement is incorrect. The amplitude at an antinode is the sum of the amplitudes of the two interfering waves, typically resulting in a maximum amplitude that is twice that of each individual component wave, assuming they are of equal amplitude.
  4. All the particles between two adjacent nodes vibrate in phase: This statement is correct. Within the span between two nodes (in one loop of the wave), all the particles move in phase—they reach their maximum and minimum displacements simultaneously.
The correct and logically consistent statements, upon re-evaluation, are 2 and 4, indicating that particles on opposite sides of a node are out of phase (statement 2), and particles between nodes are in phase with each other (statement 4).
Answer Option: A: 2 and 4 are correct.
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Consider the following statements-In a stationary wave1. All the parti...
Understanding Stationary Waves
Stationary waves, also known as standing waves, are formed by the interference of two waves traveling in opposite directions. Let's analyze the given statements one by one.
1. Frequency of Particles
- All particles in a stationary wave perform simple harmonic motion.
- However, the frequency of the particles is the same as that of the component waves, not four times. So, this statement is incorrect.
2. Phase Difference Across Nodes
- Particles on opposite sides of a node do indeed vibrate with a phase difference of π (180 degrees). This means they move in opposite directions at any instant. This statement is correct.
3. Amplitude at Antinodes
- At an antinode, the particles have maximum amplitude, which is equal to the sum of the amplitudes of the two component waves, not equal to either of them. Hence, this statement is incorrect.
4. Particles Vibrating in Phase
- All particles between two adjacent nodes vibrate in phase, meaning they reach their maximum displacement simultaneously. Thus, this statement is correct.
Conclusion
Based on the analysis:
- Statements 2 and 4 are correct.
- Statements 1 and 3 are incorrect.
Therefore, the correct option is a) 2 and 4 are correct. This aligns with the principles of stationary waves in physics, making the understanding of wave behavior essential for topics like JEE.
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Consider a rope fixed at both ends under tension so that it is horizontal (i.e., assume the rope is along the x-axis, with gravity acting along the z-axis). Now the right end is continually oscillated at high frequency n (say n=100 Hz) horizontally and in a direction along the rope; amplitude of oscillation is negligible. The oscillation travels along the rope and is reflected at the left end. Let the total length of rope be I, total mass be m and the acceleration due to gravity be g. After initial phase (say a minute or so), the rope has (BLANK-1)_ wave, which is_(BLANK-2)__in nature. It results from superposition of left travelling and right traveling _(BLANK-3) waves. This resulting wave has a frequency__(BLANK-4) _ that of oscillation frequency nu, Simpy&fmensional analysis indicates that the frequency of can be of the form:___(BLANK-5)__.

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Consider the following statements-In a stationary wave1. All the particles perform simple harmonic motion with a frequency which is four times that of the two component waves.2. Particles on the opposite sides of a node vibrate with a phase difference of π.3. The amplitude of vibration of a particle at an antinode is equal to that of either component wave.4. All the particles between two adjacent nodes vibrate in phase.Of these statements-a)2 and 4 are correctb)1,3 and 4 are correctc)1 and 2 are correctd)1, 2, 3 and 4 are correctCorrect answer is option 'A'. Can you explain this answer?
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