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Tags math puzzles , physics , structural engineering , trigonometry

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Old 14th June 2011, 09:11 AM   #1
Oystein
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Our hammock: The forces to hold it

Hi folks,

my girl friend and I recently bought a hammock for my rooftop terrace. It didn't come with extra rope, so my girl friend went to a home improvement store later that day and bought some. I told her not to be too thrifty and buy something strong, as the forces pulling at the ropes of a hammock can be larger than the weight it carries.

The rope had a tag that says "up to 100kg"

Now: Is that strong enough? I will now present my physics and math and kindly ask anybody with too much time if I am doing it wrong:


Here is my hammock, for the moment without anyone in it, just a couple of pillows:


The purple line is horizontal, the green and red line vertical, the blue line is along the rope.

My take on the physics is this: The force vector pulling on the rope is a combination of forces pulling vertically (the pure weight of the hammock and its content) and horizontally (the other end of the hammock).

To make it easier to talk about things, here is the little triangle to the left of my photo, with sides labled a, b, c and corners labelled A, B, C. The question I am attempting to answer is: Given my own mass of M=80kg, what force pulls at either rope, dependent on the angle alpha (at A)?



When alpha is 90, there is no horizontal force, and each of the two ropes carries half the weight of the hammock, M/2 = 40kg, or require a force of 40kp
When alpha is 0, the hammock is pulled so strongly horizontally that it does not sag at all. This is physically impossible, it would require an infinite frorce.
When alpha is 45, horizontal and vertical forces are equal. The vertical force, which I shall call a must always be M/2, so the horizontal component then is also f(M)/2, and the total force, according to the Pythagorean theorem, is sqrt((f(M)/2)2+(f(M)/2)2) = f(M) * sqrt(2)

Now some trigonometry:
a = tan(alpha) * b or b= a/tan(alpha)
c = srqt(a2 + b2)

With a=f(M)/2 (f(M) being the force that holds a weight of M against gravity; for example, to hold a weight of 80kg, you need a force of 80kp), we know now that
b= (f(M)/2)/tan(alpha)
c = sqrt((f(M)/2)2 + (f(M)/2)/tan(alpha)2) or f(M)/2) * sqrt(1 + 1/tan(alpha))

So at an angle of 30 away from horizontal, the ropes of my hammock experience a force of

40kp * sqrt(1 + 1(tan(30)) = 40kp * sqrt(1 + 1/0.577...) = 40kp * sqrt(1 + 1,732...) = 66.115...kp

Here a table for several other angles:
angleforce (kp)
0#DIV/0!
1305,39
7120,96
8113,95
9108,18
10103,32
1199,15
1587,01
2077,43
3066,12
4556,57
6050,24
9040

Seems like that rope that is ok up to 100kg can hold me in hammock down to an angle of 11.
In the photo above, it appears that the angle is slightly larger than 20, and that would increase once I get into the hammock, to maybe 25-30. Looks I am on the safe side, pulling with only 66-77kp on the ropes.


So folks, is my physics and math correct? And how would the shape of the hammock itself or its occupant play a role here?

Last edited by Oystein; 14th June 2011 at 09:13 AM.
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Old 14th June 2011, 09:19 AM   #2
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I can't help with the math, but the rope looks like polypropylene. As someone who put many antennas up for ham radio, I can tell you that poly decays very quickly in sunlight and breaks. Also, if you should happen to jump into the hammock, you will stress the ropes beyond 100kg, and, if your girlfriend happens to jump in it with you, down you both go onto the tile. I think I'd get some chain or good hemp rope from the hardware store.
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Old 14th June 2011, 09:31 AM   #3
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The math looks right to me, but there are some additional things to consider.

First, while the shape of the hammock and your body has no direct effect, it can have an indirect effect. For calculating the tension in the rope, only the angle of the rope matters. But the shape of the hammock, as well as how you sit in it, can affect that. In particular, if you sit in it, the ropes may take a different angle than when you lie in it.

But there's a more important issue than that. In actual use, you want the rope to be able to hold considerably more than your weight. You want the rope to able to be able withstanding not just sitting or lying in the hammock, but also jumping into it. And for that, I'd suggest you need at least a factor of 2 to be on the safe side. Your current setup doesn't have that. The rope itself probably has some margin beyond its rating, but you don't want to depend on that.

But the good news is that you don't need to buy new rope to get that. From the picture, it's clear that you've got plenty of rope. So if you loop it from the rail to the hammock, you'll double the strength. Better yet, loop it twice and you've got a factor of 4, which should be enough for any amount of jumping. You should only tie it once though, so that the rope can adjust slack between the loops. That way you can be sure that the load is shared between the loops. If you tie the rope multiple times, then the loading won't be even.

Edit: I have no experience with sun damage on ropes so I didn't think about it, but make sure to pay attention to Olowkow's warning above.
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Last edited by Ziggurat; 14th June 2011 at 09:33 AM.
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Old 14th June 2011, 10:08 AM   #4
Oystein
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I knew I wouldn't have to wait long for smart replies. Thanks, pals!

Yes, the rope is some cheap plastic, so yes, it will grow old in the sun.
No, I do not "jump" in the hammock, and we don't lay in it together, it's too simple and flimsy for that.
I thought of looping, but having the rope double does not cut the strain in half, this only happens if they can move against each other with no or little resistance, as in a pulley. But that is still better than nothing.

I am also experimenting with knots. One end is fixed with a carabiner, on the other I prefer a rolling hitch, which depends on some starin to hold. Doubling the rope might give me too little pull.
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Old 14th June 2011, 10:30 AM   #5
Ziggurat
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Originally Posted by Oystein View Post
I thought of looping, but having the rope double does not cut the strain in half, this only happens if they can move against each other with no or little resistance, as in a pulley. But that is still better than nothing.

I am also experimenting with knots. One end is fixed with a carabiner, on the other I prefer a rolling hitch, which depends on some starin to hold. Doubling the rope might give me too little pull.
If you loop the rope, you can tie it against itself with a fisherman's knot or a double fisherman's knot (aka grapevine or double grapevine).
http://www.layhands.com/Knots/Knots_...FishermansKnot
You can still clip the carabiner to the loop, but you don't need to tie the rope to the carabiner. In fact, by simply clipping it on, the loops will be able to slide against the carabiner enough to balance the load. And you only need the loop to be around the railing, so you won't have the issue that you do with the rolling hitch.

Edit: the only drawback is that it's a bit of a hassle to tie a fisherman's knot with a rope significantly longer than needed, so you'd probably want to cut the rope shorter.
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Last edited by Ziggurat; 14th June 2011 at 10:32 AM.
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Old 14th June 2011, 11:14 AM   #6
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About the math, it's not the tangent function you want, it's the sine function.

If you're 80 kg and the rope angle is 20, then each side would be holding a vertical component of 40 kg, so the force vector would be 40/sin(20) = 116 kg (yes, I'm treating "kg" as force, but the rope is spec'd like that).

Then consider that the force will increase greatly as you plop down into the hammock, but I think you should be OK with four runs of rope on each end.

Are you sure the railing can support that kind of force pulling on it?




ETA: why would the rope angle change significantly when you're in it? I don't think it would.


ETA2: I think you were using the tangent function to find the horizontal component of force, then using the Pythagorean theorem to calculate the hypotenuse force. That's doing it the hard way. The sine function gets you there directly.
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Last edited by CurtC; 14th June 2011 at 11:21 AM.
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Old 14th June 2011, 12:51 PM   #7
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Oh, my goodness. Just buy a stronger rope.





Seriously though, I am impressed.
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Old 14th June 2011, 01:13 PM   #8
Oystein
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Originally Posted by CurtC View Post
About the math, it's not the tangent function you want, it's the sine function.

If you're 80 kg and the rope angle is 20, then each side would be holding a vertical component of 40 kg, so the force vector would be 40/sin(20) = 116 kg (yes, I'm treating "kg" as force, but the rope is spec'd like that).
Originally Posted by CurtC View Post
ETA2: I think you were using the tangent function to find the horizontal component of force, then using the Pythagorean theorem to calculate the hypotenuse force. That's doing it the hard way. The sine function gets you there directly.
Hmm sin?
Let's see: http://en.wikipedia.org/wiki/Trigonometric_functions
In my triangle,
- sin(alpha) = a/c
- tan(alpha) = a/b

so...
c = a/sin(alpha)
By jolly! You are right! I am learning things about the good old trig funcs I never knew before!

But where is my mistake? Hmmm... Maybe I should do that on good old paper, not in the edit window of some web forum...

Originally Posted by CurtC View Post
Then consider that the force will increase greatly as you plop down into the hammock, but I think you should be OK with four runs of rope on each end.
I always ease myself in gently ^^

Originally Posted by CurtC View Post
Are you sure the railing can support that kind of force pulling on it?
Yeah, pretty sure, that's heavy duty and does not yield the least bit to any force I can apply with my weight and muscles.

Originally Posted by CurtC View Post
ETA: why would the rope angle change significantly when you're in it? I don't think it would.
Mainly because rope and hammock are elastic and stretch under load. Think of a guitar string when you pull on it.
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Old 14th June 2011, 01:31 PM   #9
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Originally Posted by Oystein View Post
...
But where is my mistake? Hmmm... Maybe I should do that on good old paper, not in the edit window of some web forum...
...
Ah! I dropped a square somewhere, should have been

c = a * sqrt(1 + 1/tan(alpha)2)


Which, by the way, apparently means
sin() = 1 / sqrt(1 + 1/tan()2)
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Old 14th June 2011, 02:16 PM   #10
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If the rope breaks when you are on it, make sure you do not fall very far. Also test it out first.
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Old 14th June 2011, 02:40 PM   #11
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Originally Posted by Oystein View Post
I thought of looping, but having the rope double does not cut the strain in half, this only happens if they can move against each other with no or little resistance, as in a pulley. But that is still better than nothing.
The frictional loss of that rope around that post and the rings will be very nominal if you set it up properly. If it were my hammock, and intended to relax in it, I would double loop it and use a single knot along the lines of what zig has suggested. Then pull on the hammock ring to equalize the lengths before you put a serious load on the rope. If they are nearly equal before loading they will share the load rather well.

Truth be told, even then I wouldn't relax very much. Sorry, I spent too many years on the top of rocks laying out climbing rigs to be comfortable with the safety margins you are dealing with here. Even if the fall is only a foot or so. We use retired climbing rope or chains for our hammocks. The relaxing is very good.
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Old 14th June 2011, 06:32 PM   #12
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Bear in mind that knots produce stress concentrations that reduce the strength of the rope, sometimes significantly. There's some numbers here...

http://www.caves.org/section/vertica.../knotrope.html
http://en.wikipedia.org/wiki/Knot#Strength

Up here we have universal health care. If you're paying your own medical bills, or a percentage of them, go buy some decent rope. Look for a climbing store, sailing supply store, etc... get polyester. It will last longer in the sun than nylon.

ETA: Whoops, from the "posting" screen I couldn't see you were in Germany. By all means, let the government pay for your broken tailbone.

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Old 14th June 2011, 06:59 PM   #13
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Just judging by the apparent thickness of the rope, I'd put a couple cushions BENEATH the hammock for WHEN (not IF) the rope breaks to soften the impact on your backside.

Seriously -- invest in a thicker rope. You don't want to have to remember to "ease" into your hammock. You want to flop in it. Plus, it's just a matter of time before your girlfriend wants to curl up with you in it.

Half-remembered line form Julie Haggardy in some movie: "I lost my virginity in a hammock -- you had to have good balance."



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Old 15th June 2011, 03:29 AM   #14
Oystein
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Hehe thanks for all your input. Yes, I will get new rope soon.


Originally Posted by Beanbag View Post
Just judging by the apparent thickness of the rope, I'd put a couple cushions BENEATH the hammock for WHEN (not IF) the rope breaks to soften the impact on your backside.
As a matter of fact, I do that already

Originally Posted by Beanbag View Post
Seriously -- invest in a thicker rope. You don't want to have to remember to "ease" into your hammock. You want to flop in it. Plus, it's just a matter of time before your girlfriend wants to curl up with you in it.

Half-remembered line form Julie Haggardy in some movie: "I lost my virginity in a hammock -- you had to have good balance."



Beanbag
Balance is the main reason why I don't jump into it but get in carefully: I fell out twice when I moved too quickly
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Old 15th June 2011, 11:48 AM   #15
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Just get some 550 cord (also known as paracord or Type III nylon), it's wonderful and cheap.
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Old 15th June 2011, 03:49 PM   #16
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Originally Posted by excaza View Post
Just get some 550 cord (also known as paracord or Type III nylon), it's wonderful and cheap.
Paracord is really the wrong thing for this task. A decent 1/2" three-strand polyester will run about a dollar a foot and he only needs (at most) about 15 feet.
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Old 16th June 2011, 04:33 AM   #17
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1/2" polyester? He's trying to hold up a hammock, not a pickup truck.
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Old 16th June 2011, 04:51 AM   #18
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Erm...as a math-illiterate, assuming you calculate everything right with some safety margins built in, your weight will still be partly supported by the materials of the hammock itself. If that rope is enough to support you, wouldn't it also support the hammock? You need more of what you have - or if that won't hold you you need to sue the makers of the hammock. Was there any disclaimer on the packaging? "Not intended for use as a hammock", that sort of thing?
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Old 16th June 2011, 08:51 AM   #19
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Originally Posted by Dr. Keith View Post
I would double loop it and use a single knot along the lines of what zig has suggested. Then pull on the hammock ring to equalize the lengths before you put a serious load on the rope. If they are nearly equal before loading they will share the load rather well.
Especially if the rope is a little stretchy like he said, the lengths should share the load well.
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Old 16th June 2011, 11:16 AM   #20
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Originally Posted by excaza View Post
1/2" polyester? He's trying to hold up a hammock, not a pickup truck.
Oh yeah, the capacity would be totally overkill on 1/2" polyester (7500lb tensile strength, according to New England Ropes). What I was thinking about was more along the lines of ease of handling and untying the knots after strain was put on them.
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Old 17th June 2011, 02:41 AM   #21
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Alright folks,

the main purpose of this thread was to check my math with regards to structural properties of the situation, and, thankfully, that was sorted out (my idea was correct, but I did it the hard way and made an error of arithmetic somewhere).

I did buy 4 meters of better rope yesterday. Don't know how to translate this to your terms ("550 rope", "7500lb tensile strength" etc.). It had two numbers: One apparently for tensile strength, which I forgot, but it was a 4-digit thing in kg; another number was for - well, I don't know, there was a picture with rope horizontally, and a weight of 145kg pushing down vertically. I figured I would be totally on the safe side with this, and also it had a lable of "UV: +", meaning it is well suited for use in sunlight.

Next thing is to figure out which knots are best. My girl friend used to teach climbing, so she is the expert on knots, we'll do that later today.

Thanks to all who weighed in!
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Old 17th June 2011, 02:51 AM   #22
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Originally Posted by Oystein View Post
I knew I wouldn't have to wait long for smart replies. Thanks, pals!

Yes, the rope is some cheap plastic, so yes, it will grow old in the sun.
No, I do not "jump" in the hammock, and we don't lay in it together, it's too simple and flimsy for that.
I thought of looping, but having the rope double does not cut the strain in half, this only happens if they can move against each other with no or little resistance, as in a pulley. But that is still better than nothing.

I am also experimenting with knots. One end is fixed with a carabiner, on the other I prefer a rolling hitch, which depends on some starin to hold. Doubling the rope might give me too little pull.
How far are you from the ocean, or a lake large enough to have a yachting port? If convinient go there and get a good rope (there will be an accessory shop). They run into breaking strengths of hundreds of kilos, and they can stand the sun for years. They are pretty expensive, but you can buy them per lenght, so you don't have to get any surplus. Also, they look nice.

Hans
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Old 17th June 2011, 03:09 AM   #23
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Originally Posted by MRC_Hans View Post
How far are you from the ocean, or a lake large enough to have a yachting port? If convinient go there and get a good rope (there will be an accessory shop). They run into breaking strengths of hundreds of kilos, and they can stand the sun for years. They are pretty expensive, but you can buy them per lenght, so you don't have to get any surplus. Also, they look nice.

Hans
3 hours to the Dutch coast. We have an excellent outdoor equipment store in town where I would expect to find quality climbing equipment (ropes made to hold humans in the worst of conditions) and possibly some sailing equipment, too.
But I already got some good enough rope now. It's still only a hammock, and I won't be using it in stormy weather
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Old 17th June 2011, 03:40 AM   #24
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I had the railing come off the wall which turned out to be made of compressed guano.
Yours looks sturdier.

So, what happened? Broken coccyx?
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Old 17th June 2011, 04:18 AM   #25
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Originally Posted by Oystein View Post
Next thing is to figure out which knots are best. My girl friend used to teach climbing, so she is the expert on knots, we'll do that later today.
Just follow the one simple rule: Can't tie a knot? Tie a lot!
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Old 17th June 2011, 06:28 AM   #26
Oystein
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Originally Posted by lemurien View Post
I had the railing come off the wall which turned out to be made of compressed guano.
Yours looks sturdier.

So, what happened? Broken coccyx?
I am currently running observations on my girl friend in the hammock. She did three palsteks, and I attached the final rolling hitch. So far everything's fine
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Old 17th June 2011, 10:19 AM   #27
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Originally Posted by Oystein View Post
I am currently running observations on my girl friend in the hammock. She did three palsteks, and I attached the final rolling hitch. So far everything's fine
Looks like that is a "bowline" to English speakers - good Climbing knot...

Dave
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Old 17th June 2011, 11:10 AM   #28
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I believe we'll need photo evidence of girlfriend in hammock to determine if this was a valid experiment.
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Old 17th June 2011, 11:19 AM   #29
excaza
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Who needs rope?
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Old 18th June 2011, 05:21 AM   #30
Oystein
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Originally Posted by excaza View Post
Uhm...
Quote:
With about three rolls of duct tape, two large wooden dowels, and 20 feet of rope...
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Old 18th June 2011, 05:22 AM   #31
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Originally Posted by CaveDave View Post
Looks like that is a "bowline" to English speakers - good Climbing knot...

Dave
Correct. Somehow, Palstek didn't occur to me as a German words. Sorry 'bout that.
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Old 18th June 2011, 05:24 AM   #32
Oystein
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Originally Posted by JSFolk View Post
I believe we'll need photo evidence of girlfriend in hammock to determine if this was a valid experiment.
I was talking about observations, not experiment
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Old 18th June 2011, 06:16 AM   #33
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Originally Posted by alfaniner View Post
Oh, my goodness. Just buy a stronger rope.
That might be my approach, too.


Quote:
Seriously though, I am impressed.
Me, too. Reminds me of why I come to this place.
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Old 18th June 2011, 11:01 AM   #34
lemurien
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I hope that these pauses in observation are filled with passion, in the hammock or elsewhere...I remember how exciting it was to make love on the balcony, watching the people down on the road...

Why is it people get old and stiff in all the wrong places?
I hate it!
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Old 19th June 2011, 04:15 AM   #35
Oystein
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Originally Posted by Sideroxylon View Post
...
Me, too. Reminds me of why I come to this place.
Yes, absolutely. To pass idle time and rest (when it isn't warm on the terrace), my girl friend watches daily soaps and show bizz news, and I read and chat on JREF. She thinks I am wasting my time...


#
Originally Posted by lemurien View Post
I hope that these pauses in observation are filled with passion, in the hammock or elsewhere...I remember how exciting it was to make love on the balcony, watching the people down on the road...

Why is it people get old and stiff in all the wrong places?
I hate it!
Hmm well, two problems:
1. Hammock is too small for the two of us
2. Four of the neighbouring houses have rooftop terraces at about the same level. It's nice for a little small talk and sharing experiences with potted plants, not so nice for relaxed love-making
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