Is the "clack" sound from Newton's Cradle periodic?

[StevenJacobs990]

They certainly do sound periodic from observation. But is there a particular formula that proves that sound from Newton's Cradle is periodic?

 

[phinds]

I think of periodic as a uniform period. Isn't it the case that with Newton's Cradle the period decreases very slowly but constantly as the balls to go less and less high and return more and more quickly?

 

[russ_watters]

I think of periodic as a uniform period. Isn't it the case that with Newton's Cradle the period decreases very slowly but constantly as the balls to go less and less high and return more and more quickly?

I would think not since each end is half a pendulum. That assumption and a zero transit time for the impulse is all it takes to show it is simple harmonic motion. And even woth the delay it would still be periodic.

 

[phinds]

I would think not since each end is half a pendulum. That assumption and a zero transit time for the impulse is all it takes to show it is simple harmonic motion. And even woth the delay it would still be periodic.

I thought the small amount of energy lost as heat due to the impact would change that. I guess your assumption of zero transit time makes it an ideal case with no lost energy so we are describing slightly different things, yes?

 

[russ_watters]

I thought the small amount of energy lost as heat due to the impact would change that. I guess your assumption of zero transit time makes it an ideal case with no lost energy so we are describing slightly different things, yes?

No, zero transit time is different from lost energy. With zero transit time you get a decaying amplitude sine wave (the decaying amplitude is the lost energy, whether it is in the impact or in air resistance). Including the transit time just gives you a little pause each cycle. This is why pendulums make good clocks: period is independent of amplitude.

 

[DarkBabylon]

I thought the small amount of energy lost as heat due to the impact would change that. I guess your assumption of zero transit time makes it an ideal case with no lost energy so we are describing slightly different things, yes?

And this is why we need to be a bit more advanced and talk about differential equations.
In simple words, the friction would make a function that multiplies the sinusoidal function. No term from the friction will be present inside the sin function itself.
What this means: The period does not change, only the amplitude as russ_watters said.

 

[phinds]

No, zero transit time is different from lost energy. With zero transit time you get a decaying amplitude sine wave (the decaying amplitude is the lost energy, whether it is in the impact or in air resistance). Including the transit time just gives you a little pause each cycle. This is why pendulums make good clocks: period is independent of amplitude.

And this is why we need to be a bit more advanced and talk about differential equations.
In simple words, the friction would make a function that multiplies the sinusoidal function. No term from the friction will be present inside the sin function itself.
What this means: The period does not change, only the amplitude as russ_watters said.

Thanks guys.

 

[phinds]

@StevenJacobs990, the thread got a bit derailed by my misunderstanding but the answer is yes its periodic. As to the exact equation with exact values, that will depend on the parameters of the pendulum.

 

[StevenJacobs990]

@StevenJacobs990, the thread got a bit derailed by my misunderstanding but the answer is yes its periodic. As to the exact equation with exact values, that will depend on the parameters of the pendulum.

But what proves that the sound is periodic? I'm not asking for the exact equation with exact values.
I want to know whether the period stays constant regardless of the decreasing amplitude (the height the ball on each end reaches, which decreases over time because of energy loss).

So I thought of this pendulum equation T = 2(pi)(sqrt of (L/g)). Does this prove that Newton's cradle is periodic even over time?
Because according to this equation, length of the string that holds the mass is the only thing that affects the period, not the amplitude.

 

[Bystander]

https://en.wikipedia.org/wiki/Pendulum_(mathematics)

 

[DarkBabylon]

Since Newton's cradle has a periodic motion, the sound must also come periodically. Every time the mass passes the lowest height it can get, you'd get the sound.
However the time period of the sound would actually be half of the pendulum's.

 

[Fervent Freyja]

The period does not change, only the amplitude as russ_watters said.

Would the period change depending upon the number of spheres in the cradle? I seem to hear 3 spheres in motion 'clanking' more frequently than when 5 are in motion?

 

[Bystander]

actually be half of the pendulum's

... which is actually a function of the amplitude; not a strong function admittedly but still ...

go less and less high and return more and more quickly

period is independent ofn amplitude.

... , sorry Russ, but phinds was correct.

 

[DarkBabylon]

Would the period change depending upon the number of spheres in the cradle? I seem to hear 3 spheres in motion 'clanking' more frequently than when 5 are in motion?

We actually had a lab on the subject and I measured it again in some videos of Newton's cradle. The frequency does not change, even if the masses were coupled in case of a symmetric motion (the two coupled masses move together in the same way).
This is why in physics we don't use "seem" to draw conclusions. We do however allow 'seem' to be used as a reason why we are testing something, to raise a question as you did. Once we test it with proper equipment, we sometimes see our intuition was wrong. We are awful at processing data beyond survival.

Edit: After some derivation using Newton's laws of motion I came to a conclusion that the masses should move together. Using this knowledge you can convert the mass into a mass twice bigger on a single string (or two parallel to one another so the mass would only move on one axis but for calculation we assume one) so the only effect you would have is in conservation of momentum.
The frequency does not change.