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.

File size:
15 KB
Views:
70
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