Poll

5 votes (26.31%)
2 votes (10.52%)
1 vote (5.26%)
3 votes (15.78%)
7 votes (36.84%)
4 votes (21.05%)
2 votes (10.52%)
5 votes (26.31%)
2 votes (10.52%)
4 votes (21.05%)

19 members have voted

Wizard
Administrator
Wizard 
Joined: Oct 14, 2009
  • Threads: 1191
  • Posts: 19912
January 5th, 2019 at 9:55:09 PM permalink
A 100 meter rope is suspended from the top of two 50-meter poles. The lowest point of the rope is 10 meters from the ground. How far apart are the poles?

This is a tough one, so I'll give some hints.


The equation of a dangling rope from two points at the same height is a catenary.

The general equation of a catenary is a*cosh(x/a) - a = a * (exp(x/a)-exp(-x/a))/2 - a.

I'll give you these derivatives:

cosh'(x) = sinh(x)

sinh'(x) = cosh(x)

And finally this one nobody should be without:

cosh^2(x) - sinh^2(x) = 1

That should be enough to get you to the answer.


As usual, please put answers in spoiler tags and not too much cheating.

Don, I'd kindly ask you to give the rest of the forum 24 hours to earn a beer this time.

The question for the forum is what is the answer?
Last edited by: Wizard on Jan 6, 2019
It's not whether you win or lose; it's whether or not you had a good bet.
teliot
teliot
Joined: Oct 19, 2009
  • Threads: 37
  • Posts: 1956
January 6th, 2019 at 6:36:50 AM permalink
I hope you don't mind a bonus question Mike --

Is the answer different on the moon? (gravitational force = 1/6-th that of Earth).
BleedingChipsSlowly
BleedingChipsSlowly
Joined: Jul 9, 2010
  • Threads: 16
  • Posts: 835
January 6th, 2019 at 8:03:52 AM permalink
Quote: teliot

I hope you don't mind a bonus question Mike --

Is the answer different on the moon? (gravitational force = 1/6-th that of Earth).

I don't have time to scrape the rust off what I remember about solving differential equations, but I'll take a guess at the bonus question.

The briefs I read about the seminal work done on this (Leibniz, Huygens, et al.) reference a hanging chain, not a rope. Also, none of the material makes allowance for differences in gravity, e.g. examples on mountain tops as opposed to sea level. I am going to infer from this the answer to the bonus question is "yes" for a rope which will stretch under load. If a chain had been used for the problem, my guess for the bonus question would have been "no."
ďYou donít bring a bone saw to a negotiation.Ē - Robert Jordan, former U.S. ambassador to Saudi Arabia
OnceDear
Administrator
OnceDear
Joined: Jun 1, 2014
  • Threads: 43
  • Posts: 3855
January 6th, 2019 at 8:33:31 AM permalink
I can't remember how to do the maths. But I do remember that the curve is called a 'catenary'
When was the last time you used that in casual conversation?
If you are enjoying the game, you're already winning.
Wizard
Administrator
Wizard 
Joined: Oct 14, 2009
  • Threads: 1191
  • Posts: 19912
January 6th, 2019 at 9:43:24 AM permalink
Quote: teliot

I hope you don't mind a bonus question Mike --

Is the answer different on the moon? (gravitational force = 1/6-th that of Earth).



I actually have wondered this for about 40 years. Since a catenary (why doesn't my spell checker like that word?) is noteworthy because there is equal gravitational force at every point, if that is the correct way of putting it, I would say it is the same on the moon.
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Administrator
Wizard 
Joined: Oct 14, 2009
  • Threads: 1191
  • Posts: 19912
January 6th, 2019 at 9:44:58 AM permalink
Quote: OnceDear

I can't remember how to do the maths. But I do remember that the curve is called a 'catenary'
When was the last time you used that in casual conversation?



I look for any excuse to steer conversations towards such topics. Probably explains, in part, why I can never hold a woman's attention for more than a minute.
It's not whether you win or lose; it's whether or not you had a good bet.
unJon
unJon
Joined: Jul 1, 2018
  • Threads: 8
  • Posts: 931
January 6th, 2019 at 11:02:50 AM permalink
Quote: Wizard

I actually have wondered this for about 40 years. Since a catenary (why doesn't my spell checker like that word?) is noteworthy because there is equal gravitational force at every point, if that is the correct way of putting it, I would say it is the same on the moon.



Well itís been more than 20 years since we derived the catenary in Calc 3 in college, but I thought it wasnít just equal gravitational force but also that being equal to the force caused by the tension of the rope. If thatís right, then less gravity means the balance would be with less rope tension, meaning the y-axis minimum would be higher if gravity was less.

But I could have that totally wrong.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Wizard
Administrator
Wizard 
Joined: Oct 14, 2009
  • Threads: 1191
  • Posts: 19912
January 6th, 2019 at 11:10:41 AM permalink
I think we have to assume zero tension in the rope for it to be a catenary. Perhaps I should have said chain. If there is tension, the curve will take a form between at catenary and a parabola.



I think it could be said the shape of such arches work because the gravitational pressure is equally distributed along the curve. In other words, there is no weakest link.

p.s. You guys have just under 11 hours before Don swoops in and snatches another beer.
It's not whether you win or lose; it's whether or not you had a good bet.
ChesterDog
ChesterDog
Joined: Jul 26, 2010
  • Threads: 6
  • Posts: 736
January 6th, 2019 at 11:23:19 AM permalink
Quote: unJon

...then less gravity means the balance would be with less rope tension...



Instead of thinking about the effect of the local value of g, we can ask, "Do both a string catenary and a heavy chain catenary of the same length take the same shape?"
Wizard
Administrator
Wizard 
Joined: Oct 14, 2009
  • Threads: 1191
  • Posts: 19912
January 6th, 2019 at 11:41:44 AM permalink
Who are the four people with an X gender here?

Gender 'X': New York City to add third gender option to birth certificates
It's not whether you win or lose; it's whether or not you had a good bet.

  • Jump to: