# I Damped Oscillations

#### rugerts

A pendulum with no friction/resistance/damping (i.e. in a vacuum) will swing indefinitely.
Does a pendulum with damping effects ever truly stop oscillating? That is, does the graph tend to infinity or actually reach a value of 0, i.e. the equilibrium position?

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#### anorlunda

Mentor
Mathematically, the oscillations decrease asymptotically.

But in real life, there are always nonlinearities, so yes it stops.

But also in real life, we have micro earthquakes, so no it doesn't.

It's just a matter of definition. How small do the oscillations get before you call them "zero?"

#### rugerts

Mathematically, the oscillations decrease asymptotically.

But in real life, there are always nonlinearities, so yes it stops.

But also in real life, we have micro earthquakes, so no it doesn't.

It's just a matter of definition. How small do the oscillations get before you call them "zero?"
I think I see what you mean. If it's small enough (whatever that is), it will be effectively zero in the real world. In "math modeling world", it never really reaches 0. Like you said, though, if we're really dealing with non-linearities and the imperfectness of the real world, can we ever say it actually reaches zero? Or is this just not measurable and therefore an invalid (and more philosophical) question?

#### anorlunda

Mentor
Ican we ever say it actually reaches zero?
Don't you see? This is not a math or a technical question but rather a language question. How do you define actually zero?
How small? Lasting for how long?

#### jbriggs444

Mathematically, the oscillations decrease asymptotically.
In the case of a damped pendulum we would typically consider linear damping -- a retarding force that is proportional to the pendulum's velocity. With small (under-damped) linear dampening the motion of the pendulum will turn out to follow a sine wave multiplied by a decreasing exponential.

e.g. $x(t) = cos(t) e^{-t}$ (with constants added to get the right phase, frequency and decay rate).

You can find more than you may have wanted to know here: http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

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#### rugerts

Don't you see? This is not a math or a technical question but rather a language question. How do you define actually zero?
How small? Lasting for how long?
Zero = no movement

#### Dr.D

It all depends upon the kind of damping present. Viscous damping will never totally stop the motion, but dry friction (Coulomb) damping will bring the motion completely to rest.

#### hutchphd

It all depends upon the kind of damping present. Viscous damping will never totally stop the motion, but dry friction (Coulomb) damping will bring the motion completely to rest.
Is it not a consequence of the fluctuation-dissipation theorem that viscous damping will necessarily be accompanied by fluctuations and that these will cause the motion to reverse (at least for an instant) after a finite time?

#### rude man

Homework Helper
Gold Member
This is an example of how the real world is infinitely more complex than any physics you'll ever learn can explain fully. Everything you learn is approximation. So it's best you accept those limitations; there's plenty left for you with which to be challenged!

#### Mister T

Gold Member
Does a pendulum with damping effects ever truly stop oscillating?
Of course it does.

That is, does the graph tend to infinity or actually reach a value of 0, i.e. the equilibrium position?
The graph is part of the model used to describe the behavior of the pendulum. If your model involves a damping factor proportional to the speed then that model will predict an infinite amount of time for the pendulum to stop. By the way, at some point the velocity and the position predicted by the model will need to be known simultaneously with an uncertainty that's less than the Heisenberg Uncertainty. In other words, the model fails outside of its limits of validity.

"Damped Oscillations"

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