How Are Simple Harmonics Related to Circular Motion?

[WPMcB1997]

Hello PF,

This is my first EVER thread in this website.
Excuse me for my bad english. I'm a junior high student from Thailand. Yep, you heard it, Junior high. (Studying in a Math-Sci Program)

Back to the topic.
Can you give me some basic ideas on simple harmonics? I've just finished circular motion and it seems to me that simple harmonics isn't as "simple" as it's supposed to be. Anything similar to Circular motion?
At least just give me a guidance on where to start please.

Please excuse me if I'm posting something wrong or posting in the wrong section.
Help would be very appreciated.

 

[Pythagorean]

harmonics is a very general word, you might want to be more specific. In the meantime, I think the string of a musical instrument is the best intuitive demonstration of harmonics:

http://en.wikipedia.org/wiki/Harmonic_series_(music )
(not to be confused with the mathematical harmonic series)

you strike a string; it vibrates as a whole (the fundamental) but it also has harmonics that ring at 1/2 the string, 1/3, 1/4, and so forth, each with diminishing amplitude. Helicopter blades have similar harmonics; as they slice through the air, the vibrate at several nodes.

 

[WPMcB1997]

Ahhh...I'm sorry...wrong word

I meant the simple harmonic motion...my bad.

 

[Pythagorean]

circular motion can be a type of simple harmonic motion. Many things in the universe can be modeled as simple harmonic motion. Things that go back and forth display simple harmonic motion as well as things that go around in a circle again and again. They can be driven by all kinds of classical forces: gravity, springs, electrodynamics, tension.

Have you read the wiki?
http://en.wikipedia.org/wiki/Simple_harmonic_motion

 

[technician]

WPM:
If you have studied CIRCULAR MOTION and feel confident with the maths then you should be able to cope with SHM.
All of the equations in SHM can be obtained from the equations in circular motion.
If you imagine a point on the edge of a disc rotating at constant angular velocity then I hope you know the following equations for the circular motion ;
speed of point v = ωr
Centripetal acceleration a = rω^2 or v^2/r Directed towards the centre of the circle.

If you can look at the disc EDGE ON then the point will be going from side to side and this motion is SHM. Without going through all of the maths some things stand out about SHM.
The Amplitude of the SHM is the radius of the circle
The maximum velocity is when the point passes through the mid-point... this is the velocity (speed) of the point on the edge of the circle seen edge on.
The maximum acceleration is at the max displacement and this is the circular motion acceleration
So in SHM max velocity v = ωA (A is amplitude)
max acceleration a = Aω^2
This means that max KE is when point passes through mid-point = 0.5mv^2 =0.5m(ωA)^2
This also means that the max PE is at the max displacement and must also be = 0.5m(ωA)^2
If you go back to the circle and draw a radius from the centre to the point, call the angle between the x-axis radius and the radius to the point ∅ = (ωt) then you may see where the expressions
x = A Cos(ωt) a = Aω^2Cos(ωt) and v = ωASin(ωt) come from
Hope this helps
You may find these equations written as
x = A Sin(ωt) a = Aω^2Sin(ωt) and v = ωACos(ωt)
This happens if you decide to call the angle between the radius to the point and the y direction ∅
This depends on the textbook you use !

 

[WPMcB1997]

Thanks technician...that helps a lot!
That applies to both springs and pendulums?

 

[technician]

WPM
Applies to absolutely everything that does SHM...in every case you can imagine a circle rotating at the angular frequency viewed EDGE ON

 

[WPMcB1997]

Thank you very much!

 

[technician]

Once you are used to the idea you will say "there is no such thing as SHM...it is only circular motion viewed edge on"
All of the equations in SHM can be found from the equivalent circle.
Good luck...SHM is considered to be difficult by most students.
Contact me again if you need help.