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Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

Posted June 01, 2014 12:00 AM
Pathfinder Tags: challenge questions

This month's Challenge Question: Specs & Techs from GlobalSpec:

Two cylinders of equal length and radius are set at the same height on a ramp and allowed to roll to the bottom. The first is a solid aluminum cylinder. The second is a hollow lead cylinder with an inner radius slightly more than 3/4 of its outer radius. Assuming the frictional effects are negligible, which cylinder reaches the bottom of the ramp first?

And the answer is:

The solid aluminum cylinder will reach the bottom first.

The key to this problem is that the cylinders are rolling down the ramp. In this case the masses of the cylinders are unimportant, for the same reason that they would be unimportant if the cylinders were just dropped from the same height (neglecting air friction). What matters in this problem is the difference in the moment of inertia of a solid cylinder as compared to a hollow cylinder (cylindrical shell). This difference leads to different accelerations for the respective cylinder's center of mass.

Center of Mass Acceleration of the solid cylinder:

Where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp.

Center of Mass Acceleration of the hollow cylinder:

Where g is the acceleration due to gravity (9.8 m/s^2) and θ is the angle of the ramp.

Since the acceleration of the center of mass of the solid cylinder is greater than the acceleration of the center of mass of the hollow cylinder, the solid cylinder reaches the bottom of the ramp first.

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#1

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 12:55 AM

The aluminum one; its rotational inertia is less, hence it accelerates a bit faster.

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#10
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 1:07 PM

I think you have the correct answer and a GA from me.

Torque is a factor, and for a solid cylinder I = 1/2 MR2.

Where a ring of equal mass is I = MR2.

If both mass and radius are the same, then:

mgh = 1/2 mv2 + 1/2 Iw2

Note, that I is either 1/2 MR2 or MR2 depending on the body.

To answer another ambiguous question that others have made a comment on, I think negligible means just enough friction for rolling to commence.

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#11
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 1:34 PM

Distinguishing between air resistance and surface friction might have helped. Since rolling was mentioned, I read it as air resistance being negligible. In the case of sliding, europium is of course correct.

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#13
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 3:57 PM

I remember enough of these problems from high school and college to know that the crux of the problem is concerned with fundamental classical mechanical physics. Going beyond that was the realm of advanced degrees and theses.

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#30
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:36 AM

"... I think negligible means just enough friction for rolling to commence."

In other words, "'negligible' can mean anything we want it to mean so long as it supports our foregone conclusion." That, in a nutshell, is what you are saying.

How can friction be considered 'negligible' when its presence or absence is fundamental to the outcome?

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#39
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 9:43 AM

Well, re-reading the question states the cylinders are allowed to "roll", so sliding should not be a factor and the impact of any other forms of friction are to be discounted (negligible).

Sometimes a rose is just a rose.

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#2

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 1:49 AM

"Assuming the frictional effects are neglible..." then *no torque* will be exerted on the cylinders and they will *slide* to the bottom without rotating, both reaching it at the same time.

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#40
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 11:48 AM

I would have liked to have been in your physics class in school. I bet you gave the teacher a hard time!

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#42
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 2:59 PM

My high-school Physics teacher, Dr. Norton, was a retired physicist from Sandia National Laboratories. It was he who gave us a hard time! Dr. Norton was very rigorous and, were he here, he would have ripped this Challenge Question (and most of the Challenge Questions posted here) a new ____hole. I'm merely continuing his long and notorious tradition of questioning hidden assumptions, identifying experimental biases, not leaping to foregone conclusions and ... running against the grain.

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#84
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 4:42 PM

OK, I'm a little guilty too. I remember the geometry teacher saying he would give an 'A' to anyone who could draw a equilateral right triangle. I brought him one drawn on a ping pong ball.

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#86
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 5:49 PM

Now that's just plain wicked.

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#87
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 10:41 PM

Did you get the deserved 'A'?

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#85
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 4:49 PM

I expect that if the Challenge Questions specified every detail precisely, it would give away the solution.

In this case, they could have left out any mention of friction. The difference in the moment of inertia to weight ratio between the two cylinders would be more significant than differences in rolling friction.

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#3

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 3:34 AM

It sounds a bit like Galileo at the Leaning Tower of Pisa.

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#4
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 7:47 AM

Hi Kris,

Yes, and more than a bit.'

I'll bet you a fiver that the 'Correct answer to be revealed later this month' will in fact be maximally incorrect. Wanna bet?

"Negligible" friction, they said; friction, but at such a reduced level as to have no influence on the outcome of he experiment. That's what 'negligible friction' in this context means to me. They went out of their way to mention this. Even so, I'm wagering that the forthcoming 'correct answer' will implicitly assume non-negligible friction in contradiction to the original problem statement.

The framers of this problem are expecting those cylinders to rotate. You can tell by the way they worded the question. They're expecting the arrival times to be different due to the difference in angular inertia, as pointed out by Tornado.

But in order for angular inertia to even be relevant in this problem, there must be rotation, and if there is rotation, there must be a cause - those cylinders are not going to rotate by themselves as if by magic, and doing so would violate the conservation of energy not to mention a few other physical laws.

Something has to exert a torque on those cylinders to get them to rotate. That something is friction. Non-negligible friction. Non-negligible to such an extent that it fundamentally alters the outcome of the experiment.

If the 'correct answer' assumes even one iota of rotation, the 'correct answer' will be dead wrong. Without friction, the only possible motion is translational. Were we to conduct this experiment in real life, somehow reducing real friction to a negligible degree, those cylinders will slide to the bottom of the ramp, without rotating, and will arrive at the same time, a la Galileo.

Make that a tenner.

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#5
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 9:24 AM
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#23
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 6:17 PM

rofl!

xkcd has got to be the best website on the planet, bar none.

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#6
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 9:39 AM

A couple of pints ain`t too much, but the airfare will bust yer ****`s :) Meet you in Etherville next week, and bring your wallet !

Without friction, the answer on this will be a lol. I`ll not care - a fairly good seafood place and a few pints slips down fairly good .

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#22
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 6:15 PM

Hell, I don't need an ambiguous, assumption-packed Challenge (hic) Question for a motive. And seafood? I'll bring two wallets! That or you fly to N'awlins for some kick-arse crawfish etouffé, dirty rice, gumbo, fried gator and liquid sustenence of your choice. Mint julips ain't bad for starters. You'll be a Rajin' Cajun in no time.

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#26
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 9:55 PM

That sounds really good either way- don't let anyone else know, the bar/eatery hasn't been built that's big enough if everyone comes along, but we must sort this out some time. That Southern style grub has me salivating !

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#28
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 11:57 PM

Some great Nawlins restaurants featuring cajun, creole, Spanish creole, haute cajun and contemporary. Most are in the French Quarter except for Commander's Palace (bottom) which is in the Garden District. [click on image to go to website]

.

Antoine's are expecting you. A great place to launch your culinary expedition.

.

Arnaud's

.

Redfish Grill

.

Borgne

.

Commander's Palace

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#8
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 11:19 AM

"The framers of this problem are expecting those cylinders to rotate."

Really? Maybe because the 'problem' clearly states "...allowed to roll to the bottom."

I think this is another 'done in one' challenge. Tornado provided the obvious correct answer. Perhaps if the challenge also included a powerful magnetic field, or other complicating factor, the answer would have required more thought.

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#12
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 3:26 PM

Yes, they certainly are allowed to roll and so we have the implicit demand that they must, and so, in one fell swoop, Physics flies out the window.

They're also allowed to reach the bottom of the ramp at different times, and therefore they must. The only difference is that the framers stopped just short of saying so, and so the real Challenge Question is "Forget what we wrote. Who can guess our hidden assumptions most accurately?" Physics has little to do with it. Scientific method even less so.

If this had been an early 20th-century CR4 Challenge Question about the difference in the speed of light as measured by two arms of a cross-beam interferometer, the obvious 'correct' answer (cough) would be the speed of the lumeniferous aether.

Fortunately for Science, Michaelson & Morely got it 'wrong.'

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#14
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 3:57 PM

Michelson, Morley, luminiferous.

These challenge questions are often framed incompletely or ambiguously.

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#16
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 4:08 PM

Michelson, Morley, luminiferous.

Thank you. Yes, I noticed also - 16 minutes after posting.

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#19
In reply to #14

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 4:48 PM

"Challenge Questions ... incomplete... framed ambiguously."

It is surprisingly difficult to frame such questions to exclude hidden assumptions. That fact that the assumptions are *hidden* makes it so. The problem is further compounded by the fact that natural language is, by its very nature, ambiguous.

What I think would have made a better Challenge Question here would be to pose the question as framed along with the 'correct answer', and ask

1) "What hidden assumptions are we making that lead to this particular solution?"

2) "If one were to take our problem definition at face value, how would the outcome be different, or would it?"

3) "If different, why different? If not, why not?"

For my part, I think that would stimulate some very interesting dialogue!

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#25
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 9:37 PM

I completely agree. I offered my scenario of the flattened lead pipe sliding on the ramp while the aluminum rod rolls just to point out that a valid but unlikely scenario meets the stated conditions and produces the antithesis of the expected result. This is another poorly phrased problem in giving us a unique answer.

However, the very same ambiguity of the given conditions (negligible friction effects) has sparked a great discussion here at CR4 that has been missing lately. IMHO the parsing of this ambiguity and the subsequent variety of valid results from how one interprets this ambiguity has been far more informative than just getting a simple dry result answer.

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#27
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 10:30 PM

This is exactly why I presented an alternative solution: to highlight the hidden assumptions implicit in the problem definition.

It is obvious from the wording (and the title) what the *expected* 'correct' answer should be, but the answer, were we to take the problem definition *strictly at face value,* that is, as being complete unto itself and containing all the information necessary to arrive at a correct solution, is something altogether different.

The problem definition, as worded, is in fact unambiguous. What *is* ambiguous are the hidden assumptions the author made whilst formulating the definition. What these assumptions are will be made clear when the author reveals what he or she considers to be the 'correct' answer.

I might also point out that hidden assumptions, biases and foregone conclusions are the arch-nemesis of Good Science.

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#117
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/06/2014 12:47 AM

It's very much like Galileo's experiment, except that in an experiment like this, where they're rolling down an incline (barring slippage and other complications) a hollow cylinder will always flag behind a solid one regardless of any other difference between them. They could differ in radius, mass, length, density and/or composition. It doesn't make a bit of difference; as long as they both have uniform radial density, the solid cylinder will always win.

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#118
In reply to #117

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/06/2014 11:12 AM

rofl - I am still going with the pic you posted of the cylinders in an `x` formation.Some variations come to mind, but that is a brilliant way of busting how a question is phrased.

I said nothing about how Gailileo dropped his balls, so to speak. The thing was leaning over before even completed, and does any of the sources from history specify, here to mess with argumants, surely MR G would be near the top of the list.

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#7

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 10:22 AM

Might the low strength of the lead make a deformation in the lead tube. Thus more friction will be needed for the lead tube to start rolling. So if the aluminum cylinder does roll with only negligible friction (an oxymoron scenario but given) and the lead tube slides due to a deformation and insufficient friction for rotation then the lead tube should reach the bottom of the ramp first since no energy or momentum is going into rotation.

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#9

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 12:24 PM

The frictional effects are negligible, but not zero...I might add that the lead cylinder is not likely to be 100% balanced, and will have a wobble.....so the aluminum cylinder will arrive at the bottom first.....

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#17
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 4:17 PM

"The frictional effects are negligible, but not zero"

More accurately, "The frictional effects are negligible - except where they're not"

"I might add ..."

We must be very careful not to add to (nor subtract from) the problem definition. When we do so, we are no longer answering the same question but a different question altogether; one that wasn't asked.

Just my two cents' worth.

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#18
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 4:22 PM

As ever, it`s all very much in the wording. This one will be a hoot when the `official` answer is given.

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#20
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 5:30 PM

"Forget what we wrote. Who can guess our hidden assumptions most accurately?"

This accurately describes every test I've ever taken.....

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#21
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 6:04 PM
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#15

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 4:04 PM

The solid cylinder. The ratio of moment of inertia to weight is less for the solid cylinder than the hollow one.

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#24
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/01/2014 6:24 PM

What about the effects of eddy currents resulting from motion within the Earth's magnetic field?

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#29

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 2:27 AM

the aluminum will be the first down...if you think about two bags one filled with stone and the other with sand,,,which will be heavier?...the bag of sand of cause because less air is allowed in...hence the bag of stone will be lighter.. :)

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#60
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 10:29 AM

If the shape of the sand grains and the stone/s are the same (e.g. spheres) then the amount of air in each bag will be the same - for a big enough bag. Bag of sand would have more rolling hysteresis so stones reach bottom first - Pom kit wha.

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#66
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 3:44 PM
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#82
In reply to #29

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 10:28 AM

Unless you just bag up one big rock

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#107
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 7:04 PM

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#31

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:45 AM

Who said the ramp had a bottom?
Or that the cylinders started at the top?
What if their ends are touching, they will stay together.. ah, but there's no friction. It's what we call a silly question.

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Oh I said "bottom"

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#32

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 5:47 AM

They are not saying there is negligible friction, rather that its not different between the two cylinders. and imagine that it all takes place in a vaccuum so air resistance is not an issue to worry about. Now lets get out our pencils and paper and think about how we are going to solve this problem.

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#45
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:31 PM

What you wrote: "They are not saying there is negligible friction ..."

vs

What they wrote: "Assuming the frictional effects are negligible ..."

You can redefine the problem statement into a new problem statement if you like, but then your solution is no longer a solution to the original problem, but to yours.

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#46
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:40 PM

The equations were already posted in Post #10.

You will need to rearrange the equations to solve for t, but I will leave those gymnastics for others.

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#47
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 4:02 PM

Yes, but the torque on both cylinders is negligible, this being the result of negligible friction and resulting in negligible rotation. The cylinders will arrive at the bottom of the ramp, one having a negligible speed advantage over the other and so, any solution involving rotation will have a negligible advantage over one that does not. As Mr. Spock so aptly said on one STOS episode, "Any difference that makes no difference is no difference." (yes, I'm being pedantic)

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#52
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 5:49 PM

Ah, not so. I suggest an experiment.

Lie down on the top of a hill and report back when you roll to the bottom. :-)

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#48
In reply to #32

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 4:18 PM

"... imagine that it all takes place in a vaccuum"

Oh, but in a vacuum things tend to stick together (good bluddy luck getting all that Moon dust off your pants before entering the LEM). Van der Waals forces and all that, no air acting as buffer between objects to reduce the effects of electric fields at the atomic level. In a vacuum, rolling friction increases dramatically and so does stiction - a portmanteau of 'static' and 'friction.' It is why the flap actuators on high-performance aircraft intentionally vibrate at a (relatively) high frequency, even when stationary, so that the flaps can respond quickly to commands without having to be 'unstuck' first.

You don't want to put this in a vacuum. No sir! A low-pressure (~1 Torr) helium atmosphere would be far more preferable for this sort of experiment.

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#50
In reply to #48

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 5:08 PM

I think I'm going to portmanteau.....

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#51
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 5:19 PM

Ha! Hell, mate, that's a great song!

[teacher] "Bubba, please give the class an example of how you would use the word 'Timbuktu' in an anecdote."

[Bubba] "Well, me and Tim, we was out poachin' on the neighbor's ranch one night when their two girls and three friends come by in their Jeep. Well, we's got ta talkin', one thing lead to another and before you knowed it I buck three and Tim buck two."

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#33

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 7:02 AM

Nothing was said about the environment the test was conducted in. If this was done in zero gravity, none of your assumptions would be right as both cylinders would have just stayed put! LOL

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#43
In reply to #33

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:13 PM

"... none of your assumptions would be right as both cylinders would have just stayed put! LOL"

'Stayed put' with respect to what? The galactic center? Pluto? Alpha Centauri? Yuba City, California?

LOL

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#44
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 3:21 PM

Well, California is always on the move anyway.

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#49
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 4:51 PM

Not Yuba City! (with apologies to anyone here living in Yuba City, may Bog have mercy on your accursed soul ).

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#34

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:07 AM

My next Newsletter challenge lets you assume all the laws of physics are negligible.. it involves a road runner and an anvil.
Del

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#35
In reply to #34

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:25 AM

UUuummm let's see....is it an Acme anvil...?

It will always end thusly....

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#36

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:28 AM

Since friction and windage are neglected the acceleration component is proportional to gravity and ramp angle. Therefore whatever mas will reach the same speed after a time t (v=1/2a*t^2).

Both reach the bottom at the same time.

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#41
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 12:46 PM

.....but wouldn't the solid gain a head start and remain just a little in front?

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#37

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:35 AM

What was/is the angle of the ramp? At 180 degrees, the ramp itself would do little to affect the outcome and they would hit bottom simultaneously. :-)

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#38

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:59 AM

DUH! The first one gets there first!

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#53

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 8:43 PM

Let me throw another wrinkle into the ring ( I like mixing metaphors and stuff). The diameters are not given. The ramp ends when it intersects with the level ground. A large diameter cylinder does not have to travel as far down the ramp as a smaller one before it touches the ground, when the slope and the ground are simultaneously tangential to the cylinder.

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#54
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 9:24 PM

"Two cylinders of equal length and radius"

They are both the same size, so it doesn't impact the results.

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#81
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 9:06 AM

Ooops. I've done that more than once recently. I was quite sure that same diameter was not part of the question. Perhaps they are right about aging.

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#55

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/02/2014 10:26 PM

An elliptical cylinder wouldn't roll very easily. Actually, I believe any closed curve bound by two parallel planes counts as a cylinder. The word "radius" implies that the Jordan curve at the ends of the cylinder will be a unit circle. What if these are instead oblique cylinders and not right cylinders. The ends intersecting the two parallel planes could still be a circular but not parallel to the planes defined by the rolling axes. The rolling plane will intersect the cylinder in an ellipse thus making this an elliptical cylinder. The unspecified slope of the ramp may not be able to provide enough force at the assumed one g acceleration field to lift the center of mass the difference of the two axes for either cylinder to roll with the aid of any friction. Neither cylinder rolls if friction exists, if it doesn't then they slide down the friction less ramp in unison.

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#56
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 7:25 AM

What if there was peanut butter uniformly smeared over the outer surface of each cylinder?

Bet you didn't think of that?

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#57
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 7:40 AM

I will never do anything with peanut butter.

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#58
In reply to #56

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 7:40 AM

The ramp could be composed of whipped cream...in which case the lead cylinder, being the heaviest, would sink to the bottom first....

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#59
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 9:21 AM

Genius!

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#80
In reply to #56

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 7:05 AM

I don't like peanut butter. Can I have Vegemite instead?

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#61

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 11:44 AM

so higher the moment of iniertia so lower ist the acceleration of the cylinder.

the 3/4-hollow cylinder has an inertia moment of nearly 3/4 (1-81/256) of an massiv cylinder; the density of lead is nearly 4 times the density of aluminium - the massiv aluminium cylinder will reach the finish line as first.

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#62
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 12:25 PM

What if the ramp has an angle of 89°?

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#63
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 1:02 PM

.....and maybe the ramp is 24,000 miles long!

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#64
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 1:32 PM

That reminds me of some astronaut vandalism.

Two XKCD links in one thread.

That's one hell of a ramp.

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#65
In reply to #63

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 3:42 PM

And what if it has only one side? And one edge? And is made of bacon? (nom nom)

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#67
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 3:58 PM

No one has discussed warped space.

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#68
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 4:03 PM

For that you need a warp drive...

... and beer.

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#70
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 5:02 PM

Beer in a Klein bottle!

Now that's my kind of party!

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#71
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 5:49 PM

Oh god. Not another DWDT (Driving Whilst Doing Topology)

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#73
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 6:04 PM

It should have been, "Step out of the jar, sir."

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#74
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 6:05 PM

I still love these speakers. Even the follow up.

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#75
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 6:23 PM

Yes, they are over the top.

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#76
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 6:24 PM

"... and the physics are merciless."

No, they're not. Physics are very cooperative provided you're using the right physics. As awesome as these speakers are, he's doing this Auld Skool.

Too bad the bloke doesn't know about Hilbert Transforms. There are ways to efficiently coax amazingly low frequency response out of comparatively lightweight transducers. All you need is lots and lots of power and a bit of clever signal processing on a fast DSP. If I say any more my patent attorney will kill me.

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#77
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 7:08 PM

You're missing one of the key points here. These horns are meant to be efficient not power hungry.

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#78
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 7:47 PM

No, no, no. The point of my post does not relate to the efficiency nor the power consumption of these speakers, but to their *size.*

Note the leading quote, excerpted from the second pdf. This is key. He is lamenting about the limitations 'physics' imposes on the low-frequency response and how that places practical limits on speaker dimensions.

What I am asserting in my post is that *size need not be a limiting factor in low-frequency response;* that this problem can be sidestepped by taking an altogether different approach.

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#79
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 8:20 PM

There are many approaches and each has its advantages and disadvantages.

The real problem with bass is almost always the room. Some people like to use brute force and power, others skin the cat with low power and how they couple the drivers to the room.

I had a friend that would install systems in people's homes that not only had traditional speakers, but he would install a bass cabinet in the basement with the enclosure built as an acoustic piston that was anchored to the basement floor and the other half of the enclosure mechanically coupled to the floor above.

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#100
In reply to #62

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 11:34 AM

what if the ramp is a rotating cylinder and ketches the falling cylinder allways on the some angle of his surface?

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#69

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 4:37 PM

Are the cylinders side by side and both start together?.....or one behind the other?....

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#72
In reply to #69

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/03/2014 5:58 PM

Yes.

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#83

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/04/2014 11:52 AM

'...frictional effects are negligible...'

.

At least they avoided calling it 'frictionless'.

.

.

There are a number of other conditions that were not stated that would definitely affect the results. I'm not sure it is always possible to specify all the conditions that might possibly effect the outcome.

.

What about temperature? The original question states the aluminum is solid but does not comment on the stated of the lead. The temperature might be 1000° F, giving an advantage in many situations to the lead sample.

.

What about the ramp? A ramp is an inclined plane, right? So a flat plane that intersects another essentially flat ground plane, at 'the bottom of the ramp'. If that is the case, and assuming the temperature is below the melting point of lead, neither cylinder ever 'reached the bottom of the ramp', since the bottom would be a line at the sharp intersection of two planes that could not be touched by the circumference of either cylinder.

.

What about the original alignment of the cylinders placed on the ramp? This was not specified. There are numerous ways a cylinder can be placed on an inclined plane, some that allow rolling, that sill can have significant effects of how quickly the mass proceeds downhill.

.

What about the marmosets in the room? This was not specified. There are numerous ways that .....

.

.

It is good to be aware of what has not been specified. It is also good to be able to work with what was intended, even while making suggestions about improving the way in which something was asked.

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#88

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 2:30 AM

Assuming no slippage:

Solid Cylinder

Gravitational force Fg is trying to rotate cylinder around point P. The lever arm rc has a length rc = ro sin θ.

.

Torqe τ is therefore τ = Fg · rc = mS · g · ro sin θ. [1]

Find the magnitude of acceleration aS of point C parallel to the ramp:

aS = α · ro ,

where

α = τ / IS

The moment of inertia of a solid cylinder is given by Icylinder = m · r2 / 2

As the axis of rotation (through P) does not intersect the the center of mass C, we must use the Parallel Axis Theorem:

IS = Icylinder + mS · ro2 = mS · ro2/2 + mS · ro2

IS = 3 · mS · ro2/2 [2]

From [1] and [2] we can now compute aS:

aS = α · r0 = r0 · τ / IS = ro · (mS · g · ro sin θ) / (3 · mS · ro2/2)

Collecting and cancelling terms, we get

aS = (2/3) · g sin θ [3]

(psst: nary mention of mass nor radius... )

...

Hollow Cylinder

.

The moment of inertia of a hollow cylinder is given by Ihollow = m · (ro2 + ri2) /2

Where ri is the inner radius.

From the problem statement,

"The second is a hollow lead cylinder with an inner radius slightly more than 3/4 of its outer radius."

let ri = (3/4) · ro :

Ihollow = m · (ro2 + (3 · r0/4)2) / 2 = m · (25/32) ro2

As before, the axis of rotation (through P) does not intersect the the center of mass C, and so we are forced to use the Parallel Axis Theorem once again:

IH = Ihollow + mH · ro2 = mH · (25/32) · ro2 + mH · ro2

IH = (57/32) · mH · ro2

Computing the acceleration for the hollow cylinder, we get

aH = α · r0 = r0 · τ / IH = ro · (mH · g · ro sin θ) / (57 · mH · ro2/32)

..With the result

aH =(32/57) · g sin θ [4]

(psst: again, no mention of mass nor radius )

Comparing [3] and [4] :

[3] aS = (2/3) · g sin θ

[4] aH =(32/57) · g sin θ

As (2/3) is greater than (32/57), all else being equal, the solid cylinder has the greater acceleration and so we conclude that:

The solid cylinder reaches the bottom first (assuming no slippage ).

-----

I drew these images using GeoGebra.

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#89
In reply to #88

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 4:12 AM

Should you not have had:

I of cylinder = 1/2 x (M) x ((R outside) **2) - (R inside **2)).

Then substitute R inside for 3/4R outside

and solve.....

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#90
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 4:29 AM

Are you sure?

Let

I = moment of inertia,

a = inner radius,

b = outer radius,

L = length,

M = mass

and

ρ = mass density:

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#91
In reply to #89

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 4:45 AM

Let's derive this starting with the moment of inertia for a hoop with radius R:

Ihoop = m · R2.

Divide the cylinder into thin concentric hoops of thickness dR.

Density = Mass per unit volume

Density = dm / dV

where:

ρ = Density

dm = Mass of a hoop of radius R and thickness dR

dV = Volume of a hoop of radius R and thickness dR

Lets assume the height of the cylinder is h.

we have

We can obtain moment of inertia by integrating over all these hoops; in general for the first case and, in the second case, between radii a and b:

Assume the cylinder has uniform density, ie, ρ = constant

The volume of this cylinder is

and its mass M is

and since

,

the moment of inertia for hollow cylinder is therefore

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#92
In reply to #91

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 5:56 AM

Ahhh..... you are right.

I am looking up how to do the fonts for maths so I can show another way to do this.

can anyone tell me. (Like a**2)

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#93
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 6:12 AM

Click on Reply and you get the CR4 editor box with its little toolbar.

Type 'ro2' (r, oh or zero, 2)

Subscript the 'o' by highlighting it and clicking x2 on the toolbar.

Now superscript the '2' by highlighting the 2 but this time clicking on x2. It should now look like ro2.

Try clicking on the different controls to see what they do. Experiment and see what you come up with.

For the integral signs, sigmas And All That, I don't use CR4's editor. It's too limited. For math typesetting I use LaTeX.

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#94
In reply to #92

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 6:39 AM

Do NOT get LaTeX using the TeX Live link to the Windows installer. It contains a trojan. I've contacted the site admins.

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#97
In reply to #94

Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 10:16 AM

Aside from the serious point, you just said 'latex'. I'm done with this (I'd post pics, but I'd rather not be deleted)-technically it has been answered (mostly by you), and as for thoughts of latex.....

Again OT, but worth mentioning for any who wishes to read. Recent Brit news is that we are all stuffed within 2 weeks because of the GameOver Virus (which also drops a ransom virus on computers). The 2 weeks was very specific on the news. No Cr4 chat on this that I have yet seen ??????? Usually I'd PM this sort of stuff, but I'm baffled there is no thread here. Anyone is free to ask me more of what Brit news is reporting, but the current panic by government makes Y2K look like a mild fart. If anyone knows that this topic is currently up on CR4, then please let me know.

According to TV reports, all Brits are screwed within 2 weeks because of that virus. Yep, and I'm a Dutch monkeys uncle, but that is what current news is saying. Those with a sense of fun might care to check stats on where most computers infected by sleeping bot programs are. No prize for the winner.

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#98
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Re: Rolling, Rolling, Rolling...: Newsletter Challenge (June 2014)

06/05/2014 10:23 AM

Your comment is intriguing! What is this 'Dutch monkey' you speak of?

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