Why Do Negative Signs Appear in SHM Equations?

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Discussion Overview

The discussion revolves around the appearance of negative signs in the equations of Simple Harmonic Motion (SHM), particularly in the context of deriving these equations using reference circles and understanding the directionality of acceleration and velocity components.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the x component of centripetal acceleration in SHM is expressed as -rω²cosθ instead of rω²cosθ, suggesting confusion over the necessity of the negative sign.
  • Another participant points out that the definition of theta may affect the interpretation of the signs in the equations.
  • Some participants argue that the negative sign is necessary to indicate the direction of the acceleration and velocity components, as illustrated in a provided diagram.
  • There is a discussion about whether the terms vQ and aQ in the equations are vectors or magnitudes, with one participant realizing that they are magnitudes, which clarifies the use of negative signs.
  • A later contribution explains that the physical definition of SHM involves a restoring force proportional to displacement, leading to the expression F = -kx, which necessitates the negative sign to indicate direction.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and interpretation of negative signs in SHM equations. Some agree on the importance of directionality, while others remain uncertain about the implications of the signs based on their understanding of the variables involved.

Contextual Notes

There are unresolved assumptions regarding the definitions of variables and the interpretation of vector versus magnitude in the context of SHM equations. The discussion does not reach a consensus on the interpretation of the negative signs.

sylvanus
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Hi all,

I know that there are many ways to derive the equations for SHM. I'm clear on how the negative signs come out when we use derivatives; but I've a problem understanding how the negative signs come into play using the reference circle.

1. If the centripetal acceleration is a = rω2, why is it that the SHM acceleration or the x component of centripetal acceleration becomes -rω2cosθ and not rω2cosθ? The direction of the x component of the centripetal acceleration is correct, why do we need to include the negative sign?

2. Similarly, when considering the tangential velocity of a particle undergoing uniform circular motion, why is the SHM velocity -rωsinθ and not rωsinθ?

Thanks in advance!
 
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Depends on how you define theta.
 
Let's say that θ is as defined in the picture attached.
 

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    shm.png
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I guess I don't really understand the question. The x-component of the acceleration and velocity must include the correct sign to indicate direction. You can see from the diagram that the sign must be negative.
 
Doc Al said:
I guess I don't really understand the question. The x-component of the acceleration and velocity must include the correct sign to indicate direction. You can see from the diagram that the sign must be negative.

This is the part i don't understand. In the diagram, shouldn't vQsinθ be pointing towards O already? Similarly for aQcosθ as well? Doesn't the negative sign make them both point outwards of O?
 
sylvanus said:
This is the part i don't understand. In the diagram, shouldn't vQsinθ be pointing towards O already? Similarly for aQcosθ as well?
Note that vQ and aQ are magnitudes of the vectors.
Doesn't the negative sign make them both point outwards of O?
Well, check and see. For example, what's the acceleration (x-component) at θ = 0?
 
Doc Al said:
Note that vQ and aQ are magnitudes of the vectors.

Well, check and see. For example, what's the acceleration (x-component) at θ = 0?

Oh, now I get it. I'd assumed that vQ and aQ in the equations were vectors. It makes sense now if they're just magnitudes. Thanks a lot!
 
The physical definition of SHM states that when an object is displaced from its equilibrium position it experiences a restoring force which is proportional to the displacement.
This means that F = kx k is a constant (the stiffness)
x is DISPLACEMENT measured from the equilibrium position. Displacement is a vector and the expression for force must be
F = -kx
This means the acceleration is given by a = -(k/m)x
This is the place where there must be a - sign... to indicate direction
 

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