Friday, 20 April 2007

Friday's Physical Law - Round and Round

We saw a while ago that circular motion requires a centripetal force, but there is also another interesting feature that comes out from all that going round, and that is that you can do it all again (and again and again ... ad nauseum).

So what we have is a type of motion we refer to as periodic motion. That is motion that repeats itself after a period of time, and most commonly this period of time is constant (ie uniform circular motion, such as an orbit). So you start off at a certain point and then you move before coming back to where you started.

This is sort of obvious when you think about travelling in a circle. And as you can see this is a very simple sort of periodic motion.

Now since going around in a circle involves going through 360° we often describe the motion in terms of the rate at which the angle changes - the angular velocity (ω) if you will. The simplest way to think of this is, like I said, a velocity so we will consider the angle changed divided by the time taken (much like velocity is distance moved divided by time taken) and so we get

  • ω = 2π/T
But wait a minute I hear you say what does 2π have to do with angles. Well if instead of the more common degrees as a measure of angle we use radians, which are a slightly more natural way of describing angles (it all comes from the ratio of arc length to radius), then this is a measure of the angle. In fact you probably use radians with out even knowing it, when ever you say the circumference of a circle is 2πr then you have just used radians. So just to clarify:
  • 2π rad = 360°
So in other words our angular velocity is the total angle in a circle divided by the time taken to go round the circle. As we saw in the gravity post:
  • v = 2πr/T
so we can combine this with above and we see that
  • v = ωr
and we also get similar expressions for arc length (s, distance around the circumference of the circle) and acceleration (a)
  • s = θr
  • a = αr
where θ is the angle and α is the angular acceleration.

Now why is all of this remotely important, any one familiar with geometry will note that the position relative to the centre of the circle in xy-coordinates can be described in terms of the angle or indeed ω and t.
  • x = r cos(θ) = r cos(ωt)
  • y = r sin(θ) = r sin(ωt)
So in addition to being a simple periodic motion, circular motion is also a simple harmonic motion, harmonic meaning that it varies sinusoidally (like a sine or cosine wave). Now this term has a very specific meaning in physics and we will see next week just what it entails, and whether or not this is an accurate description.