Maths/Engineering-type Folk: somewhat violent question on weight balancing

WaveHopper · Jun 28, 2014

A woman, 110 lbs, stands on a high raised platform with a large hole in the floor.
The drop from the hole to the lower ground is 10 feet.
She has wrapped a length of rope three times around her chest. The rest of the rope is flung up over a beam 12 feet above the platform, and four feet from the hole. The far end reaches back to the platform floor with 5 feet spare, and its end is formed into a noose.
A man, 140 lbs, is climbing through the hole. The woman places the noose around his neck, and pushes him backwards, so that he drops through the hole.

My question - what happens mechanically to the woman? This is how I see it:
She is yanked backward and off the floor to dangle at a certain height - but how high will she be before she balances the weight of the man?
Bear in mind, the rope is through the hole, but it is suspended from the beam at an angle, and there will be friction as it meets the edge of the hole.
I don't need an exact answer - I just need to know whether
(a) I'm correct in assuming she would actually be suspended off the ground, as she is lighter than the man's weight? I'm tempted to say that her toes would be clear from the ground by few inches, but would she rise higher?
(b) she wouldn't shoot up so high that the man would reach the lower ground - she would have to rise 4 feet for this?

I actually have a diagram, but the forum editor removes spaces so it makes no sense

Hope above description is not incomprehensible.

King Neptune · Jun 28, 2014

I'd have to draw a diagram to figure the exact placements, but I think that the man would end up standing at the bottom of the hole with the rope tight at his neck, and the woman would be hanging a few feet below the beam.

Hoplite · Jun 28, 2014

King Neptune said:
I'd have to draw a diagram to figure the exact placements, but I think that the man would end up standing at the bottom of the hole with the rope tight at his neck, and the woman would be hanging a few feet below the beam.

I agree. If there's not enough friction to keep the man from falling, there's not enough friction to stop his descent at any point in time. It's not so much a problem of balancing weight, it's a problem of balancing forces. The man has an extra 30 pounds and several feet to build momentum before the woman's static force is applied.

It might be enough if the woman grabs the opening for dear life. She doesn't need to strangle the man, just apply enough force when the rope runs out to snap his neck. Assuming you want him to die.

WaveHopper · Jun 28, 2014

King Neptune said:
I'd have to draw a diagram to figure the exact placements, but I think that the man would end up standing at the bottom of the hole with the rope tight at his neck, and the woman would be hanging a few feet below the beam.

I'm glad I asked, because that is exactly what I don't want to happen. For the story to work, the man must be dead by hanging, and she frees herself with a bit of difficulty. If he reaches the floor, then I need to go back to the drawing board.
Many thanks for the input.

WaveHopper · Jun 28, 2014

Hoplite said:
I agree. If there's not enough friction to keep the man from falling, there's not enough friction to stop his descent at any point in time. It's not so much a problem of balancing weight, it's a problem of balancing forces. The man has an extra 30 pounds and several feet to build momentum before the woman's static force is applied.

It might be enough if the woman grabs the opening for dear life. She doesn't need to strangle the man, just apply enough force when the rope runs out to snap his neck. Assuming you want him to die.

Balancing forces - got it. Thanks for your time, its a great help in my avoiding a gigantic mistake.

King Neptune · Jun 28, 2014

WaveHopper said:
I'm glad I asked, because that is exactly what I don't want to happen. For the story to work, the man must be dead by hanging, and she frees herself with a bit of difficulty. If he reaches the floor, then I need to go back to the drawing board.
Many thanks for the input.

Put the platform higher off the ground and have less slack. If the man starts with the noose loose on his neck and the rope has no slack, then she can push him in, and he will fall a bit more than twelve feet, call it thirteen feet, and the woman will end up against the beam. The man probably will strangle, rather than having his neck broken. Is the rope over the beam or through an eye attached to the beam?

Bolero · Jun 28, 2014

Few thoughts and questions.

1. At the least it is going to really, really hurt the woman being yanked in the air with a rope around her chest - remember that includes breasts.

2. With the weights and forces involved, her chest could be compressed making breathing very difficult and she could pass out or even suffocate. I'd think about a quick harness round the body like a climbing harness - as in between the legs and round and up.

3. The rope could bind on itself and be really hard to undo.

4. She is now hanging at some height and has to get down. Is she going to climb the rope to the beam and undo herself there or .....?

5. Practicalities of putting a noose around his neck - he sticks his head through, she has to drop the noose over his head and down to his neck. At a minimum she must be behind him so he has the least chance to see it coming, but I'd think his hands would be up near his neck as he'd have them up for the climb.

I'm someone who pictures in detail and that is what strikes me.

WaveHopper · Jun 28, 2014

King Neptune said:
Put the platform higher off the ground and have less slack. If the man starts with the noose loose on his neck and the rope has no slack, then she can push him in, and he will fall a bit more than twelve feet, call it thirteen feet, and the woman will end up against the beam. The man probably will strangle, rather than having his neck broken. Is the rope over the beam or through an eye attached to the beam?

The rope is over the beam.
When I visualize her ending up against the beam, it's kind of cartoon-ish territory, I fear.
You've been such a great help - many thanks.

WaveHopper · Jun 28, 2014

Bolero said:
Few thoughts and questions.

1. At the least it is going to really, really hurt the woman being yanked in the air with a rope around her chest - remember that includes breasts.

Great point. I was focused on her breathing not being too restricted, but I have to consider other consequences.

2. With the weights and forces involved, her chest could be compressed making breathing very difficult and she could pass out or even suffocate. I'd think about a quick harness round the body like a climbing harness - as in between the legs and round and up.

I worried about this. I will think about it carefully.

3. The rope could bind on itself and be really hard to undo.
See 4.

4. She is now hanging at some height and has to get down. Is she going to climb the rope to the beam and undo herself there or .....?

She has a knife in her back pocket. I've written her dangling suspended, sawing away at a single spot until she falls in a heap. But that's when I thought she'd only be some inches off the ground. I've clearly got my physics wrong!

5. Practicalities of putting a noose around his neck - he sticks his head through, she has to drop the noose over his head and down to his neck. At a minimum she must be behind him so he has the least chance to see it coming, but I'd think his hands would be up near his neck as he'd have them up for the climb.
Good points.

I'm someone who pictures in detail and that is what strikes me.

Thanks for this, great help.

Bolero · Jun 28, 2014

You're welcome

I'd also consider the sort of rope and the knife. It can take a while to saw through a rope, especially one strong enough to do what you've described happening. Also, pen knife type knives can be a real swine to open.

jclarkdawe · Jun 28, 2014

There are a lot of variables here that will influence the outcome. Starting point is what type of rope? Manila is a lot less slippery then nylon, so there will be more friction to slow this down. Now let's look at diameter of the rope. Wider diameter spreads the compression force but causes more friction. And lastly is the rope run through a pulley or over a beam and what is the width of the beam?

So the guy is about four to five feet (measured at his feet) off the ground when the rope goes around his neck. This is not a long enough drop to reliably break his neck. For someone his weight, you need a drop of 8 - 12 feet to reliably snap the neck. (I can do the exact math for you if you need this.)

She pushes him on the ladder and he falls with this rope around his neck. Does he fall on his back, or sort of slide down the ladder face forward. If he falls forward, the rope tightens around his neck and he starts choking to death (different diameters of rope will produce different results).

If he falls backwards, then the effective length of his fall increases and the pressure of the rope will be against the back of his neck. Because of the angle of the rope, it's difficult to decide whether his neck would snap. My guess is probably not. Also, because the rope's pressure would not be against the front of the throat, breathing might not be compromised. However, he'd suffer some massive head and neck injury as his head continues moving after his torso stops, and then bounces back. Think whiplash in capital letters. This whiplash could be fatal from neck and/or brain bleeding.

For her to slip the rope over his head, she'll be leaning forward. Ignoring a harness, she's probably going to be drawn up by the small of her back. This position will be hard to get out of by herself without a knife, and even with a knife, would have problems. Again, diameter of the rope matters, but she's likely to have serious injuries to either her chest or abdomen. Either have a strong likelihood of being life threatening.

Where she ends up is directly in relation to how far the guy's neck falls. He could end up against the ladder with a minimal effect on her, or he could end up flat on his back with a lot more effect on her.

If she's got a knife on her, why is she standing around tied up?

Best of luck,

Jim Clark-Dawe

benbradley · Jun 28, 2014

Bolero said:
Few thoughts and questions.

1. At the least it is going to really, really hurt the woman being yanked in the air with a rope around her chest - remember that includes breasts.

2. With the weights and forces involved, her chest could be compressed making breathing very difficult and she could pass out or even suffocate. I'd think about a quick harness round the body like a climbing harness - as in between the legs and round and up.

It seems it could also break her back. I don't see the woman in any better shape than the man if that happens.

Could the woman wear a harness meant for a parachute and tie the rope to that? It seems that would be less likely to hurt her.

benbradley · Jun 28, 2014

Bolero said:
...
I actually have a diagram, but the forum editor removes spaces so it makes no sense Hope above description is not incomprehensible.

Let me see, I think there's a "code" function that will let you post (what we old-folks call) an ASCII diagram.

Here:
http://www.absolutewrite.com/forums/misc.php?do=bbcode#imgcode

Code:

Testing.
   Testing with three spaces indentation.

See if that works, putting your diagram in between [ code ] and [ / code ] tags (with the spaces removed, of course).

WaveHopper · Jun 28, 2014

Jim, thanks for the long and detailed post, with much for me to mull over.

Ben, the risk to the woman is clearly greater than I thought. And I too remember ascii-art. I'll check out the tags.

frimble3 · Jun 28, 2014

What is this 'high platform with a hole in the floor'? Is there no chance that there's some support or projection that she could tie the rope to, so that the solid object takes the weight, and not her own body? Unless this is part of your story, that he should die tied to her? Because I imagine she would be injured, even if you did the obvious thing, and made her bigger and heavier to outweigh him.

ClareGreen · Jun 28, 2014

As a straight mechanics problem, the answer's fairly obvious. When set against the real world, however, it becomes less so. What if the rope leading from the beam to the man gets forced between two planks at the edge of the hole, when pulled taut? What if the woman can find another way to add more friction, or happens to have something to grab on to?

What if she, rather than going about dropping a noose over his head and tightening it around his neck before dropping him, then cutting herself free, just cuts his throat instead?

blacbird · Jun 28, 2014

Does she weigh more than a duck?

cwack

frimble3 · Jun 29, 2014

ClareGreen said:
What if she, rather than going about dropping a noose over his head and tightening it around his neck before dropping him, then cutting herself free, just cuts his throat instead?

This is a good point. A man climbing up through a hole should be relatively defenceless for a few minutes.

Maths/Engineering-type Folk: somewhat violent question on weight balancing

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Hoplite

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Bolero

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jclarkdawe

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benbradley

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frimble3

Heckuva good sport