# SHM derivation

1. Dec 7, 2011

### sylvanus

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?

2. Dec 8, 2011

### Curl

Depends on how you define theta.

3. Dec 8, 2011

### sylvanus

Let's say that θ is as defined in the picture attached.

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4. Dec 8, 2011

### 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.

5. Dec 8, 2011

### sylvanus

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?

6. Dec 8, 2011

### 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?

7. Dec 8, 2011

### sylvanus

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!

8. Dec 8, 2011

### technician

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