I’m watching a pendulum slow down, and it has a frequency of 24 swings per minute. And it goes slower and slower, i.e. shorter and shorter distances each cycle. And eventually, it’s barely moving, but I can still see that it is, so its rate is still 24 per minute. And then it stops. What happened to that rate? I guess I’m really asking about what happens when something stops moving. Does the friction simply move it from motion to no motion? From 24 swings per minute to none? I’m not sure how to think about this. There may be less to it than I’m thinking. Any philosophy/physics people out there? xo, C.
One great thing about a pendulum – the feature that made it an ideal element for a lock – is that, to a good approximation the period depends only upon the mass of the pendulum and its length* and NOT on the amplitude of the oscillation. In other words, whether you perturb the pendulum by a little or by a lot, the period is the same. In your case, 24 swings to the minute, no matter what.
You’re saying, though, that there seems to be a discontinuous jump between 24 swings per minute and none, so when does it stop? Like all things in the real world, it happens when our simple model breaks down. When the swings get really small, the fact that you don’t have a perfect support will start to mess things up. Irregularities in the hinge at the top will perturb the motion, and it’s no longer a simple pendulum, and so the motion isn’t so simply explained. If your friction is severe enough, the motion may simply stop altogether and seize in a moment. You can’t have 24 swings per minute if you’re not moving.
But even if you had a Kater’s Pendulum balanced on mathematically perfect knife blade edges, you still wouldn’t have a simple situatioon. Random vibrations from the rest of the world will mess things up, and even if you can still move freely, your motion would transition from regular oscillation to chaos. If you isolated the vibrations, you’d get Brownian motion due to impacts from air molecules. Put it in a vacuum and you’d probably get effects from thermal effects in the metal itself, and ultimately from quantum uncertainties.
In the real world, though, the heavy footfalls of people alking by and the hum of the refrigerator going on and irregularities in your hinge will probably disrupt your perfect oscillations long before such esoteric effects come into play.
*Or, to be more precise, upon the moment of inertia and the distance between the point of suspension and the center of mass.
I am not the expert on this topic but I believe that it really does go from some motion to no motion (how else could it possibly behave?). The pendulum’s swing is eroded by friction on each swing until the final swing, when the force of gravity brings it back down to dead center and friction eats away any remaining momentum and prevents it from leaving again.
If there were no friction at all, the swinging would continue forever. With any friction at all, the only variable is how many swings it can take before all the energy is eaten away by friction. So you can’t have a scenario where the swinging continues forever and the width of the swing gets infinitesimally smaller.
This is kind of a thought experiment at this point, I don’t have any fancy equations to prove it.
I’d like to point out that even when the pendulum appears to be completely still, it still has some microscopic motion, although as CalMeacham pointed out, it’s unlikely to be the frequency it was supposed to be.
I don’t think the mass (or moment of inertia) comes into it. Here’s a link that explains that the period is equal to 2 * pi * sqrt (L / g), where L is the length of the pendulum and g is the local acceleration due to gravity.
It can be noted that if the arm of the pendulum is made of a less than perfectly rigid material, the mass will affect the length and thus the period. (But surely no one on the SDMB would ever nitpick to that extent.)
Not quite. It doesn’t depend on the mass, and this is only true for small amplitudes, so you can’t perturb it by a “lot” and have it function as a simple oscillator. To get the period to be 2pisqrt(L/g), you need to be able to assume that the sine of the angle of displacement is approximately equal to the angle itself (in radians). You can’t do that once the angle gets large.
L = length of the string
g = gravitational force
As far as I understand it, the mass of the bob has virtually no influence on frequency. That’s essentially what Galileo was trying to illustrate. The same is not true of displacement, but its effect is very small. In the largest sense, and for all intents and purposes, length is, in fact, the determining factor. The finer one measures, of course, the more prominently these other variables will seem, but in a gross sense, length is the main one, for sure.
You’re right – forgive me. It’s only g and L.
It’s effect can be very large once the displacement angle becomes “large”, with “large” being defined as the sine of the angle no longer being approximately equal to the angle. You don’t have to displace it very much for that to come into play. Then you have to take nonlinear effects into consideration, and no scientist in his right mind wants to contend with those! 
The moment of inertia most definitely factors into the pendulum motion. The standard equation for period is derived for a pendulum of point mass (but is a pretty good predictor of pendulum motion for the classic bob-type pendulum). Any pendulum with a non-negligible MOI needs that taken into account (see your link).
Quite right (and makes sense, too, when you think about it).
But it’s actually an interesting problem because
- What’s happening – the transfer of energy from kinetic to potential and back again – is actually very simple.
- BUT there is no solution for the equation of motion (so no expression for frequency)
- BUT it’s pretty easy to simulate the motion computationally (with a spreadsheet, say, and ten minutes time).
- BUT high-powered engineering dynamic analysis software often has algorithms that cause them to screw up large-angle pendulum motion suprisingly badly (by adding energy into the system).
Yes. That was a joke. Anyone who has studied physics knows how we like to linearize a problem in order to solve it. I can remember one of my professors telling us something like “hey, if you can’t linearize the equations, forget it”. We like everything to be a point particle, to act like a simple harmonic oscillator and to have no friction.
Sex for you guys must be no fun at all! :eek:
Oh, yeah, my OP. I think you’re right - the thing makes tinier and tinier swings until it pretty much stops. The event happens when the friction that slows the swing down - internal in the string, and external in the air - is greater than the energy of the swing. At that point, the thing stops. Works for me. Thanks.
This is really the important point in answering the OP. Whether you ignore the non-linearities due to a finite angle and moment of inertia and whatever or not, the amplitude of the mathematically ideal motion eventually falls below the thermal motion. If by “stopped” you mean has only thermal motion, that’s when it stops. If it isn’t “stopped” if it still has thermal motion, then it never stops.