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In a standing wave on a string rigidly fixed at both ends.
  • a)
    In one time period all the particles are simultaneously at rest twice.
  • b)
    All the particles must be at their positive extremes simultaneously once in one time period 
  • c)
    All the particles may be at their positive extremes simultaneously once in one time period.
  • d)
    All the particles are never at rest simultaneously.
Correct answer is option 'A,C'. Can you explain this answer?
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Standing Wave on a String:

A standing wave on a string occurs when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other. This interference results in the formation of nodes and antinodes along the string. The nodes are points where the amplitude of the wave is always zero, while the antinodes are points where the amplitude of the wave is maximum.

Understanding the Statements:

a) In one time period, all the particles are simultaneously at rest twice.
b) All the particles must be at their positive extremes simultaneously once in one time period.
c) All the particles may be at their positive extremes simultaneously once in one time period.
d) All the particles are never at rest simultaneously.

Explanation:

a) In one time period, all the particles are simultaneously at rest twice.
- This statement is true for a standing wave. At the nodes of the standing wave, the particles of the string do not undergo any displacement. They remain at rest. In one time period, as the wave oscillates back and forth, the particles at the nodes will be at rest twice.

b) All the particles must be at their positive extremes simultaneously once in one time period.
- This statement is not true for a standing wave. In a standing wave, the particles at the nodes are at rest, and the particles at the antinodes undergo maximum displacement. Therefore, the particles at the positive extremes (antinodes) do not occur simultaneously for all particles in one time period.

c) All the particles may be at their positive extremes simultaneously once in one time period.
- This statement is true for a standing wave. In one time period, there will be a moment when all the particles in the string simultaneously reach their positive extremes (antinodes). This occurs when the wave reaches its maximum displacement and is at a specific phase.

d) All the particles are never at rest simultaneously.
- This statement is not true for a standing wave. As mentioned earlier, the particles at the nodes of the standing wave are at rest. Therefore, there will be moments in one time period when all the particles at the nodes are simultaneously at rest.

Conclusion:

The correct answers are options 'A' and 'C'. In a standing wave on a string, all the particles are simultaneously at rest twice in one time period, and all the particles may be at their positive extremes simultaneously once in one time period.
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The French physicist Louis de-Broglie in 1924 postulated that matter, like radiation, should exhibit a dual behaviour. He proposed the following relationship between the wavelength of a material particle, its linear momentum p and planck constant h.The de Broglie relation implies that the wavelength of a particle should decreases as its velocity increases. It also implies that for a given velocity heavier particles should have shorter wavelength than lighter particles. The waves associated with particles in motion are called matter waves or de Broglie waves.These waves differ from the electromagnetic waves as they,(i) have lower velocities(ii) have no electrical and magnetic fields and(iii) are not emitted by the particle under consideration.The experimental confirmation of the deBroglie relation was obtained when Davisson and Germer, in 1927, observed that a beam of electrons is diffracted by a nickel crystal. As diffraction is a characteristic property of waves, hence the beam of electron behaves as a wave, as proposed by deBroglie.Werner Heisenberg considered the limits of how precisely we can measure properties of an electron or other microscopic particle like electron. He determined that there is a fundamental limit of how closely we can measure both position and momentum. The more accurately we measure the momentum of a particle, the less accurately we can determine its position. The converse is also ture. This is summed up in what we now call the Heisenberg uncertainty principle : It is impossible to determine simultaneously and precisely both the momentum and position of a particle. The product of undertainty in the position, x and the uncertainity in the momentum (mv) must be greater than or equal to h/4. i.e.Q. If the uncertainty in velocity position is same, then the uncertainty in momentum will be

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In a standing wave on a string rigidly fixed at both ends.a)In one time period all the particles are simultaneously at rest twice.b)All the particles must be at their positive extremes simultaneously once in one time periodc)All the particles may be at their positive extremes simultaneously once in one time period.d)All the particles are never at rest simultaneously.Correct answer is option 'A,C'. Can you explain this answer?
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