ALL ABOUT PENDULUMS

THE BASIC PENDULUM



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A basic pendulum consists of a weighted bob, and a stick, hanging from some suspension point. The pendulum is the whole thing, because when we talk about "the mass (weight) of a pendulum", we're including the mass of the stick.


[BASIC PENDULUM]
View attachment 183643
The period of a pendulum is the amount of time it takes to swing back and forth: from one side of its arc to the other, and back. Every pendulum has a regular period, which is why they're good for timekeeping. Here's the basic rule of thumb (don't worry; we'll complicate it eventually):

THE LONGER THE PENDULUM, THE LONGER (SLOWER) THE PERIOD.


View attachment 183646

The “length” of a pendulum is specified in different ways. When a manufacturer gives a pendulum length for a movement, it’s measured from the suspension point (bottom of suspension post, for example) to the tip end of the stick. But here, it means effective length: the distance from the suspension point to the center of gravity of the pendulum. The center of gravity is the point where the pendulum would balance if laid across a knife edge: the point where the amount of mass is equal on both sides. As a rough and ready estimate, that's at about the middle of the bob.

[BALANCED PENDULUM]



Clock pendulums usually have a rating nut attached to the bottom of the stick, for raising and lowering the bob, and so raising and lowering the center of gravity. Raising it makes the clock run faster. Lowering it makes the clock run slower.

[RATING NUT]
View attachment 183652

Another way to increase the rate of the clock without raising the bob is by adding mass (weight) to the stick somewhere above the bob, to raise the center of gravity. This is sometimes called “top loading” the pendulum.

[TOP LOAD]


Top loading is useful when you don’t want to shorten the pendulum for the sake of appearance, but want to make it behave like a shorter pendulm

THE COMPOUND PENDULUM

A modification of the basic pendulum is the compound pendulum, which has a second bob ABOVE the suspension point, attached to the stick with an outrigger of some sort.

[COMPOUND]
View attachment 183650

The effect of adding the upper bob is to augment the mass of the lower bob. When the lower bob is bottomed out on its rating nut, and set at maximum slowness, the added upper bob will make it run even slower.

When a rating nut is provided for the upper bob, it allows for even further rate adjustment.

[COMPOUND WITH RATING NUT]

View attachment 183649
Raising the upper bob has the opposite effect from raising the lower bob: The further the upper bob is raised, the slower it runs. This is sometimes called the “metronome effect”. A metronome is basically a compound pendulum with a long upper stick.



View attachment 183651

The further up the stick you slide the weight, the slower the metronome runs. The further down, the faster ir runs.

An important fact about compound pendulums is that, by suitably manipulating the two bobs, you can make the two adjustments cancel each other out. If you move the lower bob up a certain amount to make the rate faster, you can then move the upper bob up to cancel that effect and make it run slower again.

Understanding compound pendulums helps to understand the behavior of non-compound ones.

OKAY, IT’S NOT REALLY ABOUT THE “CENTER OF GRAVITY”

The actual important point on a pendulum isn’t the center of gravity, but the center of oscillation (center of rotation), a point somewhat below the center of gravity.


View attachment 183647
The period of a pendulum is a function of the distance from the point of suspension to the center of oscillation, rather than the center of gravity . The official definition of "center of oscillation" involves a theoretical pendulum consisting of a massless suspension rod and a massive pendulum concentrated at a dimensionless point, blah blah blah.

A definition easier to understand and easier to visualize is this: Think of an ordinary pendulum assembly (bob AND stick). The center of oscillation is the point R on the pendulum assembly (bob and rod) where, if the pendulum were detached from the suspension post and suspended from R (rather than from the other end of the stick) it would have the same period as before.

Making the inversion would turn the pendulum into a compound pendulum, with part of its mass above the (new) suspension point.

Cof O inverte]



The amount of mass now ABOVE the suspension point is the same as the mass that was formerly BELOW the suspension point. And so the period of the pendulum is unchanged. That’s what defines the center of oscillation.

MORE ABOUT THE CENTER OF OSCILLATI

Now consider the behavior of an ordinary pendulum as the bob is moved upward on the stick. When the bob is near the bottom, the center of oscillation will be a certain distance below the middle of the bob. As the bob moves upward on the stick, the center of oscillation moves along with it, and also moves closer to the middle of the bob. As the center of oscillation moves up toward the suspension point, the pendulum wags faster.

[C OF O & BASIC PEND]


The closer the center of oscillation gets to the suspension point, the shorter (faster) the period gets.

More importantly: as the bob steps up the stick toward the suspension point, the rate of change diminishes; the less effect each step will have on the period of the pendulum. Each step upward will make a smaller and smaller difference.


When the bob arrives at the exact middle of the stick, the center of oscillation coincides with the center of the bob.

[CENTER OF STICK]
View attachment 183648

If you now perform the inversion trick, and suspend it from the center of oscillation, the pendulum will spin like a propeller. No matter which end of the stick you hang it from, it will have the same period. The two ends are equal with respect to the center of oscillation.

And that's the shortest period the pendulum can have. That’s as fast as it will wag. That’s as close as the center of oscillation can get to the suspension point.

From this position of equilibrium, if the bob is moved further up the stick, the center of oscillation will migrate downward with respect to the middle of the bob (center of gravity), just as before, it migrated upward toward the middle of the bob, until they coincided. The center of oscillation is now moving away from the suspension point...and as their distance increases, the pendulum begins to wag slower.

The further the center of oscillation gets from the suspension point, the longer (slower) the period gets. And each further step will make an increasingly greater difference in the period; the inverse of its journey up to the center.


[C OF 0 & SHIFTS UP]



This means that, contrary to what you thought, it’s not true that the further up the stick you slide the bob, the faster the clock will run. There’s a point of diminishing returns; slide it up any further and the clock will slow down.

The reason is because the mass of the stick is always part of the equation. The mass of the stick below the bob must be taken into account when locating the center of oscillation.

What does this mean for the general rule?


THE SHORTER THE PENDULUM, THE SHORTER (FASTER) THE PERIOD.

Surprisingly, it doesn’t mean much. What it shows is that you don’t actually shorten a pendulum when you slide the bob up the stick. The whole pendulum, bob and stick, remains the same length.

To shorten the pendulum, and eliminate the “point of diminishing returns”, you need to chop off the stick behind the bob every time you move it up. Every time you chop it off, you reduce the mass of the pendulum, and so re-locate the center of gravity and the relationship of the center of oscillation to it. So long as that relationship remains constant, the center of oscillation will never “move back away from” the center of gravity, so there will never be diminishing returns.

The significance of all this depends on how much of the mass of the pendulum is stick, and how much is bob. If very little of it is stick (say, the stick is a very thin wire and the bob is pretty heavy) then the difference between center of gravity and center of oscillation will be miniscule, no matter where the bob is located on the stick. The two will pretty much coincide all the way up.

But if the weight is more equally distributed between bob and stick, then it can become important in establishing the period of the pendulum, and the rate of the clock.


And there will always be a point of diminishing returns along the length of the stick, no matter how light it is with respect to the bob.

—bangster

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