Friday, 4 May 2007

Friday's Physical Law - Simple Harmonic Motion

Like we saw last week simple harmonic motion (SHM) is a periodic (repetitive) motion that is descried by sine/cosine functions. There is one other requisite in the definition of SHM and that is a restoring force. What do I mean by a restoring force, well it is mostly what is sounds like, it is a force which acts to restores the system to its original or equilibrium position.

So a restoring force is at its simplest a force that acts counter to the position of the object within the system. Think of a mass on a spring - when the spring is extended it pulls the mass back, conversely if you compress the spring it pushes the mass out. So this force can be described by:

  • F = -kx
where x is the displacement and k is the spring constant, and the negative sign shows that the force counteracts the displacement.

Another example of this is a pendulum where gravity provides the force and is always trying to force the bob (the mass at the end of the pendulum) to its lowest point.

But wait if there is a force that is always pushing it back to equilibrium then how does it keep repeating its motion. Well to explain this lets look at the case of the mass on a spring (the pendulum is the same but its motion tends to be 2d and so slightly harder to explain).

If you have a mass attached to a spring and it is sitting at equilibrium and you pull it down a certain distance, then when you release it the spring pulls the mass back towards the equilibrium point, accelerating it as it goes, now as the mass gets closer to the equilibrium the force and acceleration get smaller, but the velocity gets bigger (since it is being accelerated). So when it gets back to the equilibrium x=0, so F = 0 and a = 0, however v ≠ 0 so the motion continues past the equilibrium where the acceleration now acts to slow the mass down, until it stops at the opposite point the where it started, and then accelerate it back again.

So as long as friction is small (or as well like to think for our examples non existent) then this motion will go on and on and on. This gives us our simple harmonic motion:

  • x = Acos(ωt)
  • v = sin(ωt)
  • a = -2cos(ωt)
Now so you don't look at me and ask where there heck did they come from, the first one is found in last time's discussion about circles is just the position of the mass relative toe the equilibrium, and the second two come from the definitions of velocity and acceleration, so are the change of position and velocity with respect to time. Strictly speaking these last two are the derivatives of position and velocity with respect to time which relies on calculus and gives us the instantaneous values for velocity and acceleration.

Now we saw last time that ω is related to the period of motion T. And so combining the restoring force equation and Newton's second law and the above expressions for x and a we can get:
  • F = -kx = ma
  • ω = √(k/m)
  • T = 2π √(m/k)
Now since we talk about how long it takes for things to happen with T, one other factor that is related to this is how often things happen, the frequency, f.
  • f = 1/T = ω/
The frequency of an event is the amount of occurrences in 1 second (usually measure in Hertz, Hz), it is inversely related to the period, of is something takes o.1 second to occur then it has a frequency of 10 Hz etc. OK this is only true for repetitive (oscillatory) motion other wise the frequency does not make sense.

So we can see that a mass that oscillates back and forth about an equilibrium is SHM, however circular motion does not really fit the bill since it does not have that restoring force.

Of course like the name suggests SHM is simple, but harmonic motion in general can be a lot more complicated, such things as friction can damp the motion, or something can drive the motion. Even the presence of other oscillators coupled to the first causes interesting phenomena but that is for next week.