Rockin' out, part 1. An observation and pendulum theory.

Some time back I was playing with an old-fashioned bottle opener, commonly known as a "church key".  At one point I placed the opener down so the rounded and pointed ends were facing down, touching the counter top (polished granite).  Here's a photo of the opener:

  When I put the opener down, I noticed it rocked back and forth surprisingly slowly.  Intrigued, I looked more closely at what was going on.  I saw that the opener rocked back & forth across the curved end, and in that configuration the opener was pretty stable.  It would eventually tip over if pushed over too far.

And then I tried an experiment to see if I could further increase the period of this simple mechanical oscillator.  We have some small (~1/8" square) super-magnets that are used to hold photos, coupons etc. on our refrigerator.  I stacked several of them together to raise the overall center of mass of the system, and put the stack on the opener.  Sure enough, the opener rocked even more slowly.  See below (sorry, no videos yet):

I was able to increase the period to about 1/2 second/cycle, pretty amazing considering the relatively small size and mass of the system.  Due to the relatively poor finish on the rounded end of the opener, it rocked in an irregular fashion.

I started thinking about making an "improved" version of this,  for a fun little machining project.  I did finally make one, with one false start.   But at this point, rather than just showing what I did I want to start by explaining the physics behind the mechanical oscillator, and what determines its frequency.   I will start with the pendulum, as shown below (two different positions of the sphere are shown).

The sphere is hanging on a cord of "R" length.  So what causes the sphere to swing back & forth?  Take a look at the right-hand drawing of the pendulum.  The sphere has moved over, and, due to the fact that the cord is a constant length, the sphere rises slightly.  Since the force of gravity always points down but the cord is at an angle, a restoring force appears which opposes the deflection of the sphere (this assumes that the forces due to gravity and acceleration are transmitted along the cord at angle "w").  In a dynamic situation the system exhibits a periodic transfer of energy between potential energy (due to the lift "H") and kinetic energy.  What determines the oscillation frequency?  If the cord is lengthened, for a given angle "W" "H" becomes smaller, and the restoring force becomes less.  The effect is to slow the pendulum oscillations down.  If we increase the mass, the acceleration decreases due to the relationship F = Ma where M is the mass of the sphere and a is the acceleration.  Solving for acceleration:  a = F/M.  Therefore the mass accelerates more slowly under the influence of the restoring force.  So frequency also decreases as mass increases.

Another way to look at the pendulum is as a system with a center of mass that is constrained to move around a given radius of curvature.  In these terms, the oscillating bottle opener is a similar type of mechanical oscillator.  Increasing the mass (by putting magnets on top of the the opener) decreased the oscillation frequency, just as it does for a pendulum.  We could continue to add mass until the center of mass is above the center of radius.  At that point the system would become unstable and flop over.  Unlike a pendulum.

Next time:  some implementation considerations with my rocking toy.