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We were asked a simple question on a test:

What is the maximum and minimum velocity of a particle performing an SHM?

Note here that we're talking about a generic standard SHM here. If the maximum velocity is $A\omega$ ($A$ being amplitude and $\omega$ being angular frequency), would I be right to say the minimum is $-A\omega$? The problem arises here as my physics teacher says it is 0 and that the negative sign only denotes the direction.

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I believe the problem is with the question itself, and that is leading to the confusion.

Velocity is a vector, and measures both magnitude and direction. This means that to compare two velocities, you have to specify some direction, or clarify that you are comparing the magnitudes of the two velocities. Without specifying which of the two should be applied here, I do not think the question of "maximum and minimum velocity" is a completely valid one in this context.

That being said, I believe most people would also choose the answer that your teacher chose. Without a specified direction defined as a positive velocity, it would make the most sense to compare the magnitude of each velocity when determining which is minimum and which is maximum.

The magnitude is generally more important in this context, and is what most people would focus on. Magnitudes of velocity would determine how much energy it has, whereas a negative sign would just show that it is not in the direction defined as positive; not that there is an actual negative energy the particle would have.

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  • $\begingroup$ Maybe the use of velocity here wasn't the best choice. Speed would've been the better choice. $\endgroup$
    – Akshat Deo
    Commented Nov 29, 2018 at 13:19
  • $\begingroup$ @AkshatDeo That's the problem I see. Did the question only say "velocity" and not "magnitude of velocity" or anything like that? $\endgroup$
    – JMac
    Commented Nov 29, 2018 at 13:20
  • $\begingroup$ It specifically said velocity. "Derive an expression for the maximum and minimum velocity of a particle performing SHM. " $\endgroup$
    – Akshat Deo
    Commented Nov 29, 2018 at 13:21
  • $\begingroup$ @AkshatDeo I would say the wording of the question is pretty bad then. I would have probably assumed the same answer as your teacher. $\endgroup$
    – JMac
    Commented Nov 29, 2018 at 13:40
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Your teacher is correct. You can only compare sizes of magnitude, which are never negative. The negative sign comes from the direction, not the magnitude.

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  • $\begingroup$ I think this highlights a problem with the question moreso than either answer. You can only compare sizes of magnitude; but velocity is direction and magnitude. If the question was about minimum speed, then this logic would obviously hold; but minimum velocity is less obvious, because it's not even clear what is meant by it. I don't think the question of "maximum and minimum velocity" is well defined unless you clarify a specific direction, or specify that you're comparing only magnitudes. $\endgroup$
    – JMac
    Commented Nov 29, 2018 at 13:06
  • $\begingroup$ Suppose you were to say a particle moves with respect to time as X = 6t-25. Would one not be right to say the minimum displacement is -25? $\endgroup$
    – Akshat Deo
    Commented Nov 29, 2018 at 13:17
  • $\begingroup$ @AkshatDeo I don't think it is correct. That logic is dependent on your choice of coordinate system. $\endgroup$
    – BioPhysicist
    Commented Nov 29, 2018 at 14:00
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The original question is ambiguous because it is not clear as to whether magnitudes or components are being compared although probably most students would give an answer of $A\omega$ and zero.

The question is better phrased as

What is the maximum and minimum speed of a particle performing SHM?

with an answer of $A\omega$ and zero which is comparing the magnitudes of velocity

or a form something like this which I do not really like

Assuming that a particle is performing SHM in the $\pm \hat i$ direction, what is the maximum and minimum component of velocity of the particle in the $\hat i$ direction.

with an answer of $A\omega$ and $-A\omega$, noting that if it was the $-\hat i$ direction which was specified the answer would still be the same.

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