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SHM derivation

  1. Dec 7, 2011 #1
    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 accleration 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!
     
  2. jcsd
  3. Dec 8, 2011 #2
    Depends on how you define theta.
     
  4. Dec 8, 2011 #3
    Let's say that θ is as defined in the picture attached.
     

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  5. Dec 8, 2011 #4

    Doc Al

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    Staff: Mentor

    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.
     
  6. Dec 8, 2011 #5
    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?
     
  7. Dec 8, 2011 #6

    Doc Al

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    Staff: Mentor

    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?
     
  8. Dec 8, 2011 #7
    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!
     
  9. Dec 8, 2011 #8
    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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