# A pen is attached to the wheel. Assume the pen is a sphere
**URL:** https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457
**Category:** Factual Questions
**Created:** [August 25, 2025, 2:10pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457 "2025-08-25T14:10:21Z")
**Posts on this page:** 20
**Page:** 2
### Author:  [@JWT\_Kottekoe](https://boards.straightdope.com/u/JWT_Kottekoe)
#### Post date: [August 25, 2025, 3:47pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/21 "2025-08-25T15:47:58Z")
A related fact.
The cycloid is famous in physics, since it is the solution to the Brachistochrone problem. Johann Bernoulli posed this problem: If a bead slides without friction from one point to another along a track propelled by gravity, what is the shape of the track that minimizes the time. Obviously, the shortest distance would be a straight track, but it is better to go more steeply downhill to gain speed. This increases the distance travelled, but shortens the time. The optimal track is a cycloid that falls further than the final point. In one day Newton solved this problem and showed that it is a cycloid.
[Brachistochrone Curve (Wikipedia)](https://en.wikipedia.org/wiki/Brachistochrone_curve)
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### Author:  [@Maserschmidt](https://boards.straightdope.com/u/Maserschmidt)
#### Post date: [August 25, 2025, 3:56pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/22 "2025-08-25T15:56:05Z")
> [@CookingWithGas](#):
>
> Closer to ABS(SIN)
Yes, this.
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### Author:  [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)
#### Post date: [August 25, 2025, 4:22pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/23 "2025-08-25T16:22:09Z")
That’s the good Spirograph, with the push pins and the configurable track. I got my kids the currently available Spirograph when they were young (1990s); they removed the push pins and it sucked - you just couldn’t anchor down the stationary pieces well enough without push pins.
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### Author:  [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)
#### Post date: [August 25, 2025, 4:24pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/24 "2025-08-25T16:24:08Z")
> [@CookingWithGas](#):
>
> > [@Marvin\_the\_Martian](#):
> >
> > If the pen in on the rim it would draw a sine wave.
>
> Closer to ABS(SIN)
You are correct - I was thinking only of the height of the pen above the plane. The correct answer is cycloid.
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### Author:  [@JWT\_Kottekoe](https://boards.straightdope.com/u/JWT_Kottekoe)
#### Post date: [August 25, 2025, 4:39pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/25 "2025-08-25T16:39:18Z")
The absolute value of sine has a similar shape to the cycloid, but it is much different. For example, at the cusp ABS(SIN) has a slope of one, while for the cycloid, the slope is infinite.
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### Author:  [@Robot\_Arm](https://boards.straightdope.com/u/Robot_Arm)
#### Post date: [August 25, 2025, 4:43pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/26 "2025-08-25T16:43:34Z")
> [@OldOlds](#):
>
> Would the line cross over itself at the bottom when the pen goes from moving in the direction of the bike to away from it?
A bicycle wheel wouldn’t, but a train wheel could. There’s a flange on train wheels that is larger that the rolling surface. Put the pen there and the line would form a small loop and cross over itself. When you see a train rolling forward, some small portions of it are moving backwards.
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### Author:  [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)
#### Post date: [August 25, 2025, 4:48pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/27 "2025-08-25T16:48:33Z")
> [@JWT\_Kottekoe](#):
>
> If a bead slides without friction from one point to another along a track propelled by gravity, what is the shape of the track that minimizes the time.
A straight vertical track. 😉
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### Author:  [@Robot\_Arm](https://boards.straightdope.com/u/Robot_Arm)
#### Post date: [August 25, 2025, 5:51pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/28 "2025-08-25T17:51:47Z")
If we assume a vertical cow.
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### Author:  [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)
#### Post date: [August 25, 2025, 6:08pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/29 "2025-08-25T18:08:46Z")
How do you tell when a sphere is vertical?
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### Author:  [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)
#### Post date: [August 25, 2025, 6:33pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/30 "2025-08-25T18:33:22Z")
> [@JWT\_Kottekoe](#):
>
> The absolute value of sine has a similar shape to the cycloid, but it is much different. For example, at the cusp ABS(SIN) has a slope of one, while for the cycloid, the slope is infinite.
True, I was getting more at the shape than the math.
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### Author:  [@jnglmassiv](https://boards.straightdope.com/u/jnglmassiv)
#### Post date: [August 25, 2025, 6:41pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/31 "2025-08-25T18:41:14Z")
> [@Marvin\_the\_Martian](#):
>
> ; they removed the push pins and it sucked - you just couldn’t anchor down the stationary pieces well enough without push pins
I goofed with some Spirograph drawing about two years ago. I found it quite a lot more entertaining than I remembered or expected. I got pretty good results with magnets on a steel serving tray. The key was adding a few strategic snips & folds of regular latex balloon between the gear and paper for grip. The magnets were only a couple bucks at Hobby Lobby.
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### Author:  [@Robot\_Arm](https://boards.straightdope.com/u/Robot_Arm)
#### Post date: [August 25, 2025, 6:47pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/32 "2025-08-25T18:47:26Z")
> [@Exapno\_Mapcase](#):
>
> How do you tell when a sphere is vertical?
Just keep turning it until it is.
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### Author:  [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)
#### Post date: [August 25, 2025, 7:38pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/33 "2025-08-25T19:38:31Z")
Sideways or over the top?
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### Author:  [@gnoitall](https://boards.straightdope.com/u/gnoitall)
#### Post date: [August 25, 2025, 8:03pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/34 "2025-08-25T20:03:19Z")
Yes.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [August 25, 2025, 8:37pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/35 "2025-08-25T20:37:22Z")
> [@JWT\_Kottekoe](#):
>
> A related fact.
>
> The cycloid is famous in physics, since it is the solution to the Brachistochrone problem.
And it’s also the solution to the isochron problem: A bead sliding without friction along a cycloid-shaped path will oscillate back and forth, and the time required for the oscillation will be independent of how far it’s sliding.
(this is approximately true for circular paths, which is the basis for a pendulum, but only approximately, and the approximation is best for small oscillations)
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### Author:  [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)
#### Post date: [August 25, 2025, 8:48pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/36 "2025-08-25T20:48:51Z")
Easy. When the little barber-pole thingy with the ball at its tip is at the top.
Whaddaya mean, “no”? That always worked for Bugs Bunny to locate the North Pole. 😉
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### Author:  [@markn\_1](https://boards.straightdope.com/u/markn_1)
#### Post date: [August 25, 2025, 11:22pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/37 "2025-08-25T23:22:33Z")
Ridiculous. If there were such a barber pole, it would be right in the middle of Santa’s workshop.
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### Author:  [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)
#### Post date: [August 25, 2025, 11:49pm UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/38 "2025-08-25T23:49:24Z")
Barber pole? Sounds to me like a description of a ball point pen.
Just stand it on its eraser to make it vertical.
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### Author:  [@markn\_1](https://boards.straightdope.com/u/markn_1)
#### Post date: [August 26, 2025, 1:33am UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/39 "2025-08-26T01:33:20Z")
“How about a magic trick?”
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### Author:  [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)
#### Post date: [August 26, 2025, 4:28am UTC](https://boards.straightdope.com/t/a-pen-is-attached-to-the-wheel-assume-the-pen-is-a-sphere/1022457/40 "2025-08-26T04:28:25Z")
> [@markn\_1](#):
>
> If there were such a barber pole, it would be right in the middle of Santa’s workshop.
And everyone knows the only service provided in the middle of Santa’s workshop is a dental office.
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