Q Factors

When I walk in my door when I get home, I hook my keys, which I keep on a carabiner, onto a binder clip that I've clipped onto my window sill.  Its a great way to never lose your keys.  But one thing I always notice is that when I hook it on, it swings, and every time it swings it makes a click.  This you might expect.  What always surprises me is how long the keys keep swinging.  They seem to swing for a surprisingly long time, minutes.

It always catches me off guard.  In order to explain why, I get to talk about Q Factors

The Q factor stands for quality factor.  Its a nondimensional parameter (my favorite kind) that tells you how pure your oscillator is.  

Lets back up a step.  Lots of things in the world oscillate.  Think about a swing.  If you get going on the swing and then stop rocking, you swing back and forth, back and forth, but eventually you come to a stop.   Imagine swinging on a rusted old swing set.  Now give the joint where the swing swings from a nice shot of WD-40.  You can imagine that if you repeated the experiment (get swinging to some height and then stop pumping), you'd continue to swing longer.  Why?  Because the Q factor has increased.  You're swinging on a higher quality swing.

Mathematically its defined to be

\[ Q = 2 \pi \times \frac{ U }{ \Delta U } \]

or 2 pi times the total energy stored in the oscillator divided by the energy lost in a cycle.  

But, another way to gauge the Q factor is the fact that it tells you something about how the oscillators get damped each period.  As a number it tells you how many periods need to go by for the amplitude of the oscillations to be damped by 

\[ \frac{1}{e^{2\pi}} \sim \frac{1}{535} \]

This allows you to estimate Q factors for everyday objects.  A factor of 1/535 is pretty near to my threshold for observing a lot of things.  What does a factor of 535 mean in terms of sound, one of the most common ways I interact with things around me?  Well, sound is measured in decibels, which is a logarithmic scale, where a factor of 535 in the power output by something corresponds to a change in the decibels of

\[ dB = 10 \log_{10} \frac{1}{535} \sim -27 \]

What is a decibel change of 27 mean?  Well, wikipedia tells me that a calm room is somewhere between 20 and 30 decibels, where as a TV set about a meter away is at about 60 dB.  So that tells me that if something like my keys start off making a sound comparable to the volume I set my TV at, I can listen to it until it just gets drowned out by the room and that should give me some estimate for the Q of my keys.

I'll keep you in suspense just a bit longer.  I said I was surprised how long the keys swing.  In order to put the Q that I measured in context, I'll tell you about a few other Qs of things you might have some experience with.

Most swinging things that I seem to remember coming in contact with have quality factors of about 10 or so.  Swings, or things letting a meter stick swing, stuff like that.  Tuning forks, which are built to be accurate resonators will have quality factors of about a thousand or so.  The quartz crystal in your watch, which is really supposed to be a good oscillator has a quality factor of 10 thousand or so.  One of the best Q factors achieved by man is 10^14.  

So, what was the Q factor of my keys?  I counted the times I could hear them swinging and got a count of 435.  This number isn't to be taken too seriously, but it indicated that my swinging keys have a quality factor of something between 400 and 500, which is pretty darn good for something that wasn't engineered.   That explains why it always surprises me, the keys always seem to swing much longer than I would anticipate.