TBR Physics #5.49 - standing wave's kinetic & potential energy

[whistle47]

This is from an older (~2009) version of the books. Chapter 5, #49. Passage describes a generic standing wave on a string and Figure I is just a diagram of the wave.

#49: At a particular time, the amplitude of the vibrating string in Figure I is zero. What can be said about the energy of the standing wave at this point?
A. The energy of the wave is zero.
B. The wave has only kinetic energy.
C. The wave has only potential energy.
D. The wave has both potential energy and kinetic energy.

At a node, there is no kinetic energy intuitively because that point on the string doesn't move. Mathematically, it also seems like the energy at nodes is zero for all t: http://arxiv.org/pdf/1007.3962.pdf (page 4, halfway down). But TBR's solutions claim the answer is B because of a harmonic oscillator analogy. That seems reasonable for an anti-node, but I'm not sure how it applies to nodes.

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[Cawolf]

The wave is moving at that point, but destructive interference gives it an amplitude of 0.

The entire string is always moving, unless fixed, and therefore has kinetic energy.

In regards to the paper you linked, it seems to be demonstrating that energy transfer in a standing wave is not analogous to a traveling wave - though I did see the line that says KE is zero at nodes. I am not well versed in wave mechanics, but for the purposes of introductory physics (and therefore the MCAT), there cannot be a point of the string with an energy of 0, or else conservation laws would be violated.

 

[whistle47]

The wave is moving at that point, but destructive interference gives it an amplitude of 0.

The entire string is always moving, unless fixed, and therefore has kinetic energy.

In regards to the paper you linked, it seems to be demonstrating that energy transfer in a standing wave is not analogous to a traveling wave - though I did see the line that says KE is zero at nodes. I am not well versed in wave mechanics, but for the purposes of introductory physics (and therefore the MCAT), there cannot be a point of the string with an energy of 0, or else conservation laws would be violated.

If destructive interference causes the amplitude of the wave at nodes to have zero amplitude, that means the transverse velocity at nodes is zero. It doesn't seem like nodes have longitudinal motion, because both endpoints of the string are fixed and stationary, so any point between them cannot have net motion towards either end (or, argue via symmetry of both ends). So it seems like the overall velocity at nodes is zero so the KE must be zero.

Can you explain how this would violate energy conservation?

I did come up with an alternate explanation that makes sense: unclear question wording/incorrect interpretation. The phrase "at this point" is suggestive of points on the string where the amplitude is zero (nodes), but it could also be referring to a time point rather than a physical point. In this case, the first sentence would imply that the question deals with the moment in time when the entire string is horizontal, which would make B correct.

 

[Cawolf]

There is no net translational motion, only vibrational - which is where the KE is found - the definition of a standing wave.

Personally I think the question is clear.

A wave, by definition, has energy - ruling out A.

The wave has no PE, because at zero displacement there is no stretch or tension to "store" the PE - ruling out C and D.

That leaves us with B by process of elimination. A wave has energy, it is transfer of energy. In this case it is the net vibrational motion that appears standing due to interference.

In reference to how saying the string has no KE violates the conservation of energy - if we observe a wave and can state that at zero displacement there is no PE - then if there is no KE, there cannot be a wave. This is in contrast to the observed wave.

 

[whistle47]

Perhaps we should agree that B is the correct answer, but agree to disagree on the rationale. Maybe I will come back to this question after I take the MCAT and think about it some more since it seems pretty interesting, but way beyond the scope of the exam.

Here is an entertaining exchange I found in case anyone's curious:
http://iopscience.iop.org/0143-0807/32/6/L01/pdf/0143-0807_32_6_L01.pdf
http://iopscience.iop.org/0143-0807/32/6/L02/pdf/0143-0807_32_6_L02.pdf
http://iopscience.iop.org/0143-0807/32/6/L03/pdf/0143-0807_32_6_L03.pdf

The results from the two critiques seem vaguely familiar from my physics class, but funnily enough, they both seem to contradict the basic standing wave-as-harmonic oscillator model. I suppose that's a lie-to-children...

 

[amy_k]

Kinetic energy for a string wave is determined by the up/down motion (the y component) whereas potential energy is determined by stretch of the string (the x component). Nodes are where the wave has minimum amplitude and anti-nodes are where the wave has maximum amplitude.

At the nodes, y = 0 so kinetic energy = 0 and potential energy is at a maximum.
At the antinodes, x = 0 so potential energy = 0 and kinetic energy is at a maximum.

The wave essentially is at rest at the nodes and the wave only has potential energy, making C the correct answer.

 

[amy_k]

Applying the spring-oscillation model here we would get B, since the point of "zero amplitude" for a spring is where potential energy is zero and kinetic energy maximum, but that is just not correct here for a standing wave which they are stating this is, the point of zero amplitude is where kinetic energy is zero and potential energy maximum.

 

[manohman]

bumping this because there was no consensus and I am also confused

could a berkley rep comment?

node is where there is no displacenent