Can an object be accelerating and yet -not- moving?

@MattDavis,

Normal and tangential components of acceleration. In the video you posted, the component vectors are represented as ar and at.

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@Boomslang,

Those are the "radial" and "tangential" component vectors of acceleration.
The summation of those component vectors is not constant in either direction or magnitude.
The vector that would be
a= ar + at
Is not constantly in the same direction, it is also not constantly of the same magnitude. The vector is NOT constant.

Your video actually shows this summation vector in blue.
-------------------------------------------------------------------
This is a very different situation than the one of a ball thrown in the air.
This is not "free fall".

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@MattDavis,

Thinking about it mathematically with respect to simple harmonic motion, now it makes more sense to me.
A function of a simple harmonic oscillation, for example, can be 5sin(t). Then it's possible to take s'(t), v'(t), and a'(t), which would mean that there is an infinite jerk for all simple harmonic motion that are not following cycloidal motion.
You are right, unlike free fall, there is infinite jerk involved with all simple harmonic motion but a simple harmonic motion following a full cycloidal motion.

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@Boomslang,

Let me think on that for a bit and get back to you Boomslang.... Very Happy

Looks much better right to me now, but I am a little sleepy...... I'll look with fresh eyes tomorrow. Very Happy

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@MattDavis,

MattDavis wrote:

contrex wrote:
Is that a wooden person?

Haha... yes it probably is,

So did you or did you not know his name is spelled P - l - a - n - c - k ?

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@contrex,

You should remove the Planck from your own eye...

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@engineer,

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You are way over-thinking this.

Could well be

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The position of the ball forms a parabola. It does up, it comes down.

Yes that's how I visualize it also

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The velocity….X upward to X downward…….at one second, the velocity is X/2. At three seconds it is -X/2. At two seconds it is zero…...stops moving momentarily.

Yes, theoretically for a "duration" of zero. So far we're in sync

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The acceleration never changes….always negative…….

Yes I'm beginning to catch on; what you're saying is that once I've let go of the ball the acceleration is a horizontal line representing a negative value passing through a vertical axis, with direction positive to its left and negative to its right

This is absolutely intriguing. But now I'm stuck with the notion as it passes through that latter axis, of a vertical line of zero width yet representing its value rising to zero then dropping back: The letter H on its side

One of these days gotta learn how to draw such figures digitally

I understand why you deny that crossbar in the H (where it "stops moving momentarily") but we're in an interesting territory logically and mathematically

I thank you Eng for sticking with me in my turmoil of confusion

..without becoming indignant or enraged

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@MattDavis,

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Perhaps this will help:

Probably not Matt but thank you most kindly for those links

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It is for an actual "ideal" pendulum.

Forgive me Matt but that's contradictory

Nah, I sort of know what you mean, just couldn't resist tho

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Notice first the velocity arrow does shrink to zero at the peaks.

I should think it would, if "velocity" means how fast something is going

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Notice next that the acceleration arrows are never absent (at the peaks), though they change direction.

Yea Matt, I understand that pretty well now, thanks largely to Eng's posting above which you so aptly verify herewith

Yet that creepy feeling, as I described near the end of my response, persists

I hasten to observe that that crossbar does in fact exist in a "practical" case as I've mentioned a couple of times above. But now getting theoretical it's just difficult to imagine, that crossbar has disappeared simply because its duration is zero

Especially after somebody, I forget who, maintained that acceleration precedes change in velocity (though withdrawn then possibly later reinstated, I'm still not sure), suggesting an "infinitesimal" duration at the peak; that is, our "crossbar"

…the second of two "different kinds" of zero

Mind you, I'm not arguing in my favor, only revealing the source of my confusion and the reason for tentatively reasserting my viewpoint

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If you want to expand on this even more: Look into the concept of jerk.

At first before opening the link I presumed you meant me

It's not every day……….and thank you again for your immense patience

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@maxdancona,

maxdancona wrote:

You should remove the Planck from your own eye...

It's a beam actually

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@contrex,

Con you're only beam facetious

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@Boomslang,

If you are still interested, sorry about the delay. Very Happy

Quote:

You are right, unlike free fall, there is infinite jerk involved with all simple harmonic motion but a simple harmonic motion following a full cycloidal motion.

The pendulum does have a certain level on which you can think about "simple harmonic motion", as a level of abstraction. A pendulum is a bit like a spring, but unlike a spring it is not limited to movement along only one axis.

The movement of the bob (in a pendulum) is not exactly simple harmonic motion.
If you would like here is a very witty and I think informative discussion of the rate (vector) of change of acceleration (jerk), it is not specifically about pendulums however:
http://physics.info/kinematics-calculus/

The bob (of a pendulum) does not have constant acceleration, as we can see by looking at the blue a vector in your video. The bob also does not have constant jerk. If we want to visualize this, look at how the blue acceleration vector swings clockwise and then counter clockwise. If the jerk were constant, the blue acceleration vector would have to always spin in the same direction or would have to not spin and just expand (or contract) in length.

You are right that the jerk of a pendulum is not constant, it is not strictly infinite though. It is always changing (in the case of a pendulum).

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@dalehileman,

Dale wrote:

Yet that creepy feeling, as I described near the end of my response, persists

That creepy feeling is natural Dale. You are touching upon the concept of the infinite. Our mammal brains aren't strictly designed for that sort of thing. This is the concept of the infinitesimal. It is not "natural". It requires a perspective shift.
This perspective shift is at the heart of the calculus. It is a whole other way of looking. It is an amalgamation of discrete mathematics with geometric curvature. I think you are right to feel some hesitancy. It is important to reflect back and understand that the assumptions in how we treat the infinitesimal are based on an assumption of "spacial" continuity.
Calculus is a wonderful tool for describing of ideal continuous motion of ideal continuous matter, in ideal continuous space. Your intuition is correct, this tool does have some limits.
Paths that this have taken (mathematically) is to venture out into the world of non-linear equations, or to "backtrack" to a reinvestigation of non-continuous models such as those in discrete mathematics.
It's all about complexity.

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@contrex,

contrex wrote:

So did you or did you not know his name is spelled P - l - a - n - c - k ?

If I didn't know then, I certainly know now. I hope my spelling error did not leave you too confused. My apologies if it did. Very Happy

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@MattDavis,

MattDavis wrote:

Dale wrote:
Yet that creepy feeling, as I described near the end of my response, persists
That creepy feeling is natural Dale. You are touching upon the concept of the infinite. Our mammal brains aren't strictly designed for that sort of thing. This is the concept of the infinitesimal. It is not "natural". It requires a perspective shift.

Then who is it that "designed" our mammal brains?

MattDavis wrote:

This perspective shift is at the heart of the calculus. It is a whole other way of looking. It is an amalgamation of discrete mathematics with geometric curvature. I think you are right to feel some hesitancy. It is important to reflect back and understand that the assumptions in how we treat the infinitesimal are based on an assumption of "spacial" continuity.
Calculus is a wonderful tool for describing of ideal continuous motion of ideal continuous matter, in ideal continuous space. Your intuition is correct, this tool does have some limits.
Paths that this have taken (mathematically) is to venture out into the world of non-linear equations, or to "backtrack" to a reinvestigation of non-continuous models such as those in discrete mathematics.
It's all about complexity.

It doesn't take calculus to deal with the infinetesimal, though the concepts are indeed central to it, as you wrote. Infinities are ubiquitous in our everyday concepts. The set of positive integers is infinite. The set of rational numbers (expressible as a fraction) is infinite, and, more to the matter at hand, it is easily proven that between any two rational numbers, however close they may be, there is another rational number (simply increase the denominator). Then there are real numbers, not expressable as a fraction of integers or a non-repeating decimal: they too are infinitely dense along any interval. Thus the 1 to 1 correspondence with points on a line, etc. Frankly there's not much complexity there.

Non-linear parabolic differential equations involve complexity & chaos.

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@georgeob1,

George wrote:

Then who is it that "designed" our mammal brains?

I took some liberties with respect to using the term designed in casual conversation. I do not hold the view that our mammal brains were designed (by) anyone in the personification sense of the term. You might say they are designed by selection pressures, but this is more of a "how" answer than a "why" answer. If you would like to debate epistemology/ontology distinctions I would be happy to discuss it on another forum. My goal was not to get too "deconstructionist" in my response to Dale, in effort to convey the actual points that he seems to want clarified.

My distinct impression (which I hope is in error) is that you simply want to find some fault somewhere. I am certainly fallible. If you scour the forums or search my posts you will find plenty of errors. (There is a very blatant error of mine at the beginning of this thread if that interests you).

As for the "real numbers" being simple. This is patently ridiculous look at pi (there is a vast literature available). The integers are not even simple, if you think that they are, I am sure just about any government in the world would appreciate your expertise in prime factorizations.

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Non-linear parabolic differential equations involve complexity & chaos.

This is true. These specific mathematical structures can exhibit complexity. These are not anywhere near the only ones which do. I think that you are perhaps a little confused by what exactly the difference is between choas and complexity.

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@MattDavis,

Thank you Matt for that learned exposition

Edited to add that's pronounced "lurr ned"

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@MattDavis,

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This is patently ridiculous look at pi

Reminds me Matt I've often wondered stuff like, "How are they sure it runs on," and "Suppose I wanted to find the point where there were fifteen consecutive 4's; how long would I have to search (given present digital processing rates)

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my appreciation of infinity has stuck with me since early in about 3 or 4th grade with sister Mary Consolatta. She made the relationship with 1 divided by a zero (1/0). It was many years later that I understood limits and expansions and diff equations, but my understanding of infinity was pretty much fixed in concrete

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@dalehileman,

Pi is not actually something I have much expertise in.
The "running on" without repeat is part, is a matter of pi being an irrational number.
That is to say that it cannot be expressed in terms of a ratio (a fraction| A/B etc).
There are several different proofs of pi's irrationality.
I don't have any of the proofs memorized (at least anymore). If you would like I can find one for you.
The "how long until this specific sequence appears"-type questions are very complex indeed, not an area I have spent much time on. I think (can't quite remember) that Friedrich Gauss (sp?) did some work on this type of question.

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@MattDavis,

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If you would like I can find one for you.

No but thanks Matt, I probably wouldn't understand it

1 Reply

Can an object be accelerating and yet -not- moving?

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