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TORn Amateur Symposium
Bree
Jul 21 2013, 9:09am
Post #1 of 68
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TORn AMATEUR SYMPOSIUM Day One- The Physics Of Middle Earth
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We are pleased to present the following TAS entries in the Category, The Physics of Middle Earth:
"The Physics of The Hobbit", by DwellerinDale:
The Physics of the Hobbit- Parts 1&2 by Dweller in Dale
To view an essay, please click on the appropriate link above. We hope you enjoy it as much as we enjoy posting it. The TAS is open for discussion, and any comments, questions or thought you wish to share about this particular Category pieces can be posted in this response to this thread.
Thank you for joining us in this event, and we wish you Happy Reading, TORn Brethren and Guests!
(This post was edited by TORn Amateur Symposium on Jul 21 2013, 9:10am)
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elaen32
Gondor
Jul 21 2013, 1:39pm
Post #2 of 68
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Thank you DiD, for a really interesting analysis
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As I said when the essays first came in- I really wish that you had been teaching me maths and physics at high school- it would have been so much more interesting and comprehensible. I admit that some of the maths is a little beyond me, but your verbal explanations of the maths for those mathematically challenged like me are brilliant!
It's interesting that the scene which appears less feasible, ie the falling bridge in AUJ is the one that is actually physically possible, whereas Gandalf and the Balrog is not. My initial view when watching the latter scene was concern as to how Gandalf avoided being burned by the Balrog's fire- especially when he sits on and plunges Glamdring into the Balrog. I suppose Gandalf catching up with the Balrog would be more feasible if the Balrog really had got stuck on a ledge- making his capture of Gandalf using his whip feasible, and in doing so, the Balrog dislodges himself and starts to fall again. Would that work do you think?
Coming soon!- The first TORn Amateur Symposium, starts Sunday 21st July in the Reading Room. Closing date for essay submission Sunday 14th July, but even if you don't submit, join us for some interesting discussion on some different and personal ways of looking at Tolkien's work
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DwellerInDale
Rohan
Jul 21 2013, 2:01pm
Post #3 of 68
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Hi, yes, having the Balrog get stuck on a ledge temporarily would have made the scene more physically possible, as it would have arrested his free fall. It was more dramatic just to have him fall, so as I speculated in the essay, the filmmakers didn't really need to do that, because only the time element makes the scene physically unrealistic, and they could assume that most of the audience wouldn't notice.
Don't mess with my favorite female elf.
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squire
Half-elven
Jul 21 2013, 2:32pm
Post #4 of 68
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Thanks for a thorough analysis of the Hobbit and Two Towers film scenes, in terms of real-world physics. My main impression is how well you've demonstrated that the 'reality' of feature-film photography is a double-edged sword, when rendering fantastic events for a viewing audience in contrast to a readership imagining similar events from written descriptions.
One question I have, which follows from other scenes I've seen in action films, is about the roles of instability and chaotic flow in falling scenes. I'll leave the idea of the tendency to tumble out of control in free-fall for another time perhaps, and focus on the Hobbit scene.
The main argument you give for the dwarves' survival seems to be the 'parachute effect' of their riding a flat platform to the bottom. I wonder how one can calculate the likelihood of the platform staying stable and at a normal angle to the direction of travel, thus providing the maximum air resistance? From the moment of the first break in the bridge, I would have expected the platform to tilt radically due to the differential breaking moments of the various structural members. Once one begins to give, the force on the remaining elements increases radically, leading to increasingly rapid failures in the direction of the original break: effectively dumping the load in one direction. Most bridge breaks I have ever seen photos of show that the bridge fails in a ramp-like way, with the contents sliding off before any final descent of the main structure begins, if it even does.
Then even if the bridge were accepted as initially letting loose in a horizontal mode, giving the dwarves their safe ride toward the bottom, the instability of the fall and the intersection with additional structures and resistances on the way down should, I think, have tended to tilt the bridge section in random or chaotic ways. Again, the dwarves should have been dumped into unprotected free fall, an inconvenience that would easily balance the so-called 'braking effect' of the secondary impacts. Broken bones and internal injuries, rather than black eyes, mild concussions and scratches, seems to me the least the viewers should expect even with the moderating effects of your analysis of their lighter masses and the friction of sliding rather than falling.
Finally, as dramatically as you've rendered your analysis of the film scenes, taking advantage of the fact that films offer measurable times against which to make calculations of acceleration, is it possible to compare Tolkien's written descriptions of similar scenes to see if he was being more or less realistic than the filmmakers? In other words, what about the physics of the 'real' Middle-earth?
I would take for instance Gandalf's actual fall into a lake from the Bridge of Khazad-dum as he recounts in The Two Towers (LotR III.5), and the dwarves' tumbling down the hillside in a rockslide after escaping the goblins in The Hobbit Chapter VI (which I think is the inspiration for the film's invented bridge scene). In the latter, the dwarves are saved by catching onto tree branches and trunks as they tumble into a woods on top of fast-moving rocks and boulders - I re-read the scene now and wonder if, as you put it about the similar scene in Mission Impossible, "Thorin's arm should have been ripped off at the shoulder."
squire online:
RR Discussions: The Valaquenta, A Shortcut to Mushrooms, and Of Herbs and Stewed Rabbit
Lights! Action! Discuss on the Movie board!: 'A Journey in the Dark'. and 'Designing The Two Towers'.
Footeramas: The 3rd (and NOW the 4th too!) TORn Reading Room LotR Discussion; and "Tolkien would have LOVED it!"
squiretalk introduces the J.R.R. Tolkien Encyclopedia: A Reader's Diary
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DwellerInDale
Rohan
Jul 21 2013, 3:23pm
Post #5 of 68
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Hi, thanks again for the feedback. You raise some interesting points.
The stability of the position of the bridge is a tough one. I wouldn't claim to be an expert on the aerodynamics of such a bridge, as I've never really encountered a similar problem. I think that in the film they show the back and front of the bridge breaking at the same time (perhaps they did that deliberately); given the fairly even mass distribution on top (the Dwarves weren't all clustered at one end) and those 3 tapering sublevels, and the 1000 kg weight, I think it's fairly realistic that the bridge stayed in position and did not begin to tumble. This was also helped by the fact that they never free-fall more than 3 seconds at a time. This might have been important during that last short free fall, when the bridge clears the limestone slope, but as that was only a 1-second fall, again they were fortunate that the circumstances prevented the bridge from tumbling.
For the Dwarves' expected injuries, I made the argument from scaling principles, a la Galileo's square-cube law. As someone who personally survived driving a motorcycle at 35-40 mph straight into a black Toyota SUV and walked away with only a few bruises, I'd expect that the 48 mph impact with the cavern walls might be something the Dwarves could survive without broken bones, etc. There isn't a lot of literature on this; a paper published in Serbian suggested that 3 second free falls (67 mph) are more than 50% fatal unless the fall is cushioned by something. In the film the Dwarves had the bridge and then the pile of debris at the bottom, which would help a lot.
For Gandalf's fall into a lake: the way they show it in the film, he would have died if he were a normal human. Contrary to what common sense might tell us, water is not something you want to fall into if you fall from a great height. Its incompressability will make it hard as concrete when you hit it. So Gandalf surviving the lake impact we'll have to again chalk up to magic.
Tolkien's description of the Dwarves tumbling down the hillside seems OK to me. Again it's the scaling principle- they would be like children tumbling down a hill, much less likely to be injured than full-sized humans.
Regards,
DiD
Don't mess with my favorite female elf.
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Rembrethil
Tol Eressea
Jul 21 2013, 5:06pm
Post #6 of 68
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A very well written essay. BTW did you notice that you get all of the spotlight today. LOL. It reduces a complicated subject, into an informative survey, without dumbing it down or going over anyone's head.
When I watch a movie I like to separate my critical thought ,from my fan reactions. In these different mindsets, I can enjoy a film in two ways.
I can notice all of the minutiae and mistakes in a critical manner.
Or I can suspend my belief, and just enjoy the show.
Each viewpoint has its good points, but I like to allow my imagination a bit freer reign. Perhaps the directors counted on the chaos to sell the impractical bits?
Either way, it was awesome, and this paper demonstrates an appreciation of Tolkien and inspired works.
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Brethil
Half-elven
Jul 21 2013, 5:10pm
Post #7 of 68
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To start with, glad you walked away from that impact Dweller!
To get onto my particular soapbox - my experience with brain trauma (my primary focus) is that outcomes vary based on not only speed in a restrained driver but rapidity as defined by angle of impact. So often a hideously mangled vehicle that seems to have an angled impact yields less severe deceleration injury to the highly susceptible low density brain (in the form of axonal shearing) that a slower impact which is either head on or rear end. I think (? speculating) the physics of it is a certain percentage of deflection in distributing the inertia(?) i agree that the loosely piled debris that the dwarves snd the bridge land on will absorb some of the inertia, which seems to be angled and sliding anyway by the end of the trip
Loved reading your analyses Dweller! :-)
Coming soon!- The first TORn Amateur Symposium, starts Sunday 21st July in the Reading Room. Closing date for essay submission Sunday 14th July, but even if you don't submit, join us for some interesting discussion on some different and personal ways of looking at Tolkien's work.
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DwellerInDale
Rohan
Jul 21 2013, 5:25pm
Post #8 of 68
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There must be a lot more available literature for this type of thing, given crash-dummy tests and so forth. I think this was yet another advantage of the Dwarves dropping straight downward, vs a horizontal and shearing impact, and having the bridge underneath.
Car crashes at high speeds seem to be very unpredictable. One of Dweller's uncles was involved in a head-on collision at 65 mph, was thrown through the windshield, and yet made a full recovery. So much must be due to factors such as shearing / angle of impact, as you say.
Don't mess with my favorite female elf.
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elaen32
Gondor
Jul 21 2013, 5:57pm
Post #9 of 68
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For Gandalf's fall into a lake: the way they show it in the film, he would have died if he were a normal human. Contrary to what common sense might tell us, water is not something you want to fall into if you fall from a great height. Its incompressability will make it hard as concrete when you hit it. So Gandalf surviving the lake impact we'll have to again chalk up to magic.
I always wondered about this aspect- knowing how much it can hurt even entering the water at the wrong angle from a diving board into a swimming pool. I can't remember the exact terms, but I know some of this relates to the surface tension of the water v the velocity of the falling object (oh dear- I have forgotten so much from school days ) But as Rembrethil says- one can just "enjoy the show" in movies and in the books it is even easier to ignore the feasibility of the fall. I like to think that the moment Gandalf disappears into the chasm, he becomes more Maia-like in his powers and endurance and this accounts for the catch up, staying conscious, not being burnt by the Balrog, surviving the fall into the lake etc
Coming soon!- The first TORn Amateur Symposium, starts Sunday 21st July in the Reading Room. Closing date for essay submission Sunday 14th July, but even if you don't submit, join us for some interesting discussion on some different and personal ways of looking at Tolkien's work
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elaen32
Gondor
Jul 21 2013, 6:06pm
Post #10 of 68
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I guess that with those angular hits, the impact energy is dissipated in more than one direction, compared with a straight on forward or rear shunt., just as the dwarves bridge is angled a little. One of my nieces was involved in a nasty accident when a lorry side-swiped the car she was a passenger in, at speed on the motorway. She had to be cut from the car, it was so badly damaged, but she just had some strained muscles, although the psychological trauma was worse.
The first TORn Amateur Symposium, IS NOW ON from Sunday 21st July in the Reading Room. Come and join us for some interesting discussion on some different and personal ways of looking at Tolkien's work
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cats16
Half-elven
Jul 21 2013, 6:23pm
Post #11 of 68
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I really appreciate your "in words" sections after showing the equations for any of the calculations. Admittedly, I'm not in my mathematical prime anymore. I used to be quite good at much of this sort, but I haven't done it in some time. Many things ring a bell in my head, but your process is very helpful for those like myself. 
Just a quick question regarding the free fall in The Two Towers (and I apologize if the answer is blatantly obvious and I missed it somewhere)-- Shouldn't the Balrog's falling rate have been slowed down when his whip grabs Gandalf's ankle? I know you establish that this part breaks the laws of physics, because the whip would have had to be over 700 meters long. But because we see Gandalf fighting the pull of the whip, can't we assume that the whip is taut, at it's full length, with the Balrog at the other end trying to bring him down. Then, the Balrog's falling velocity would decrease due to the pull of Gandalf's weight holding onto the bridge. I'm just wondering if that would make any difference when considering the distance Gandalf would have to fall to catch the Balrog.
I hope that makes sense, and I realize now that even if this does matter, the result would likely be negligible. (Edit: or maybe my concept of the physics for this action is just horrific, in which case please put me in my place...   )
Really like your analysis on Goblin Town, too. I second elaen by wishing you taught me for math/science in high school!
(This post was edited by cats16 on Jul 21 2013, 6:26pm)
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sevilodorf
Tol Eressea
Jul 21 2013, 7:30pm
Post #12 of 68
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or should that be wandering....tangential thoughts that popped into my head as I read...
1. Mental picture of Jackson calling the University of New Zealand physics department head .... or maybe he's listed on the credits for AUJ since that is plausible and refused to be listed on FOTR as that is implausible?
2. Balrog wings..... maybe he deployed them for a time but they aren't made for flying just gliding and collapsed... that might alter your equations.
3. Maybe the Balrog fell but grabbed a ledge or was slip sliding down initially, then used his whip which motion upset what precious balance he had achieved and down he went..... can see PJ proposing this to the physics professor with a yes hand gesture and the professor just rolling his eyes and shaking his head.
4. I am still not going to go skiing down a slope on a runaway bridge. My luck doesn't work that way. However, could bone density (assuming that dwarves' bones are more dense) account for some of their ability to walk away from all that they walk away from? Bilbo gets a pass because of his small size and Gandalf has the "I'm a wizard" thing going for him, so the dwarves need extra sturdy bones.... someone needs to tell Ori that green vegetables are a good source of calcium for strong bones.
In closing.... let me state... wonderful work...physics in practice.... now can you do the same with algebra?
Fourth Age Adventures at the Inn of the Burping Troll http://burpingtroll.com
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DanielLB
Immortal
Jul 21 2013, 7:46pm
Post #13 of 68
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A rather weird set of questions I couldn't help think of while reading Part 1.
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You are a better mathematician than me, so you might be able to alter your equations to answer my questions (or not!) 
The (apparent) strength of gravity varies according to several factors, including altitude, topography and latitude. In relation to latitude, gravity has less pull at the equator than at the poles. Though the differences are relatively small (around 0.1 m/s, I believe), would the scene have been more "realistic" had Gandalf and the Balrog fallen through Moria at the North Pole? And that leads on to my second question - you said that the main flaw in the scene was the timing element. What if Moria and the Bridge were at a considerable altitude? Acceleration due to gravity would have been less. Could that account for anything?
(I told you they were weird questions. I was also going to ask whether smoke produced from the Balrog would increase air resistance.)
And just a general comment I would like to make, I think that it is great we can use real world physics to test scenes from the films (and books). This is super geeky, thank you so much for writing this.
Coming soon! The first TORn Amateur Symposium, starts Sunday 21st July in the Reading Room. Closing date for essay submission Sunday 14th July, but even if you don't submit, join us for some interesting discussion on some different and personal ways of looking at Tolkien's work.
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DanielLB
Immortal
Jul 21 2013, 7:54pm
Post #14 of 68
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Are you suggesting that if the Company of Dwarves had in fact been a Company of Men, one would expect more than superficial injuries (and/or at least death?)
The first TORn Amateur Symposium starts this week in the Reading Room! Come and join in!
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DwellerInDale
Rohan
Jul 22 2013, 12:15am
Post #15 of 68
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In regard to the whip: the physical analysis demonstrates that the Balrog being able to grab Gandalf with the whip was impossible to begin with, unless the whip was over 400 meters long. This is due to the 12 seconds that pass between the time the Balrog begins to fall and when he grabs Gandalf with the whip. If we say OK, that really was only a second or two, then the Balrog's downward velocity may have been slowed some, but not by much--- you have a 2000 kg weight attached to an 75 kg weight, and the 2000 kg weight is falling pretty fast at that point. To try and calculate how much it might have lowered the Balrog's velocity I'd need to know things like the tensile strength of the whip-- but I don't even know what the whip was made of (does anyone?!).
Don't mess with my favorite female elf.
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DwellerInDale
Rohan
Jul 22 2013, 12:27am
Post #17 of 68
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First question: no, not really, the difference in gravity's acceleration would have made very little difference in this case. If you use 9.7 m/s^2 instead of 9.8, for example, you get a falling distance of 10.7 kilometers instead of 10.8, so the effect is negligible.
Smoke from the Balrog? I never thought of that...but again I think that the effect of his "smoke" would be negligible; he's mainly a 2000 kg mass free falling, so I can't see the smoke having much effect. He should quit smoking anyway!
The (apparent) strength of gravity varies according to several factors, including altitude, topography and latitude. In relation to latitude, gravity has less pull at the equator than at the poles. Though the differences are relatively small (around 0.1 m/s, I believe), would the scene have been more "realistic" had Gandalf and the Balrog fallen through Moria at the North Pole? And that leads on to my second question - you said that the main flaw in the scene was the timing element. What if Moria and the Bridge were at a considerable altitude? Acceleration due to gravity would have been less. Could that account for anything?
(I told you they were weird questions. I was also going to ask whether smoke produced from the Balrog would increase air resistance.)
Don't mess with my favorite female elf.
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cats16
Half-elven
Jul 22 2013, 12:36am
Post #19 of 68
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The whip really throws things for a loop here. My thought is that it seems to "extend" to the length required for its Master. Before it grabs Gandalf, it looks like it shoots high over his head first, then pulls back and grabs his ankle.
Who knows. Either way, it's nothing major. I bet PJ would be humbled, if he lurked(es?) here to see so much thought and analysis being put into the science behind the films.
So can we discuss the scientific possibility of Denethor's burning body running across the Seventh Level to...nevermind...
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Brethil
Half-elven
Jul 22 2013, 1:42am
Post #20 of 68
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Galileo's square-cube law and Tolkien's Dwarves
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Hi, yes, the effect of having a greater bone density would also have helped the Dwarves survive the impact with fewer injuries. I did not really go there in the essay because we just don't know much about this from Tolkien, but his descriptions of the Dwarves do suggest something of the kind. - DwellerinDale
In the matter of bone density: I am very intrigued Dweller, relating to the way you describe the square-cube law applying it to biological mass and proportions, it seems that JRRT's Dwarves by that definition really would have to have denser bones. They are described in the literature as stockier and (importantly) wider proportionally to Men or Elves, and thus would have greater weight to bear per linear inch of bone - so that seems, according to the models, to indicate wider bones (which I think makes sense making them tougher and more injury resistant.) So, IMHO, I think your physics confirms the word picture that we get from JRRT.
Ok, another question...
How would the Dwarves being stockier, wider for example in each horizontal dimension than other Humanoids, affect the square cube law in application? Their cross sectional muscle would be in closer proportion to their weight perhaps, giving them a greater strength to weight ratio? (Does this question make sense? I hope so.)
The first TORn Amateur Symposium starts this week in the Reading Room! Come and join in!
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Brethil
Half-elven
Jul 22 2013, 2:04am
Post #22 of 68
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Balrogs as airbags, Wizards as pancakes
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For Gandalf's fall into a lake: the way they show it in the film, he would have died if he were a normal human. Contrary to what common sense might tell us, water is not something you want to fall into if you fall from a great height. Its incompressability will make it hard as concrete when you hit it. So Gandalf surviving the lake impact we'll have to again chalk up to magic. I always wondered about this aspect- knowing how much it can hurt even entering the water at the wrong angle from a diving board into a swimming pool. I can't remember the exact terms, but I know some of this relates to the surface tension of the water v the velocity of the falling object (oh dear- I have forgotten so much from school days ) But as Rembrethil says- one can just "enjoy the show" in movies and in the books it is even easier to ignore the feasibility of the fall. I like to think that the moment Gandalf disappears into the chasm, he becomes more Maia-like in his powers and endurance and this accounts for the catch up, staying conscious, not being burnt by the Balrog, surviving the fall into the lake etc
The way that scene is filmed, Gandalf is on top of the Balrog I think - so is it an issue of him striking the water, or his deceleration factored by two things: downward force of gravity upon him in free fall AND the force of the Balrog striking the water? Would Gandalf have become sandwiched in between these two forces and made into a grey pancake? Or a 'splash' as you very funny Physicist humor suggests...?
Or would the Balrog's larger body absorb much (or enough) of the effect of the water strike, given he is tissues versus a solid, and give Gandalf a very warm airbag (against which the downward force would still push?)
The first TORn Amateur Symposium starts this week in the Reading Room! Come and join in!
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DwellerInDale
Rohan
Jul 22 2013, 2:24am
Post #23 of 68
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The square-cube law still applies, but the difference would be in their comparison to normal humans, because of the shape difference. This was one reason why I bumped up the mass estimate of the Dwarves by 25% in my analysis. If they were as wide as a normal human, for example, then with their other two dimensions being about 2/3--3/4 human values, the masses of the Dwarves would be more like 40-42 kg. With bone and muscle being proportional to the square of linear dimension, thicker bones would give them a greater strength to weight ratio.
Don't mess with my favorite female elf.
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Silverlode
Forum Admin
/ Moderator
Jul 22 2013, 2:24am
Post #24 of 68
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Thanks so much for putting the time and thought into this, DwellerinDale! Even though I know nothing about higher math, it was still a very interesting and enjoyable read. Your explanations for the laymen amongst us made it easy to follow the basic concepts if not all the details.
It's always seemed to me that the Balrog fall was shown in slow motion, to replicate the emotional impact of the event rather than to record it in real time, making it difficult to judge how quickly things like the whip strike actually happened. But if I understood correctly, even if the Balrog could have struck quickly enough to snare Gandalf with the whip, the rest of the account is impossible because a) the drag of the whip pulled by the weight and velocity of the falling Balrog would likely have jerked him right off the bridge and b) even if it hadn't, the distance the Balrog would have fallen while Gandalf clung to the edge would have put him out of reach anyway. Is that correct?
However, because it's great fun to try and analyze this, I have a question. During Gandalf's attempt to catch up with the Balrog, he goes into a dive while the Balrog seems to fall more or less flat on its back. How much would Gandalf have been able to increase his speed relative to the Balrog by reducing his own drag by diving headfirst at him, and if they started falling, say, 1-2 seconds apart, would Gandalf ever be able to catch up with him, and how far would they have to fall in order for it to happen? The Balrog did seem to bounce off the walls a few times while Gandalf was falling straight down, which I assume would have slowed him down a little.
And I also laughed out loud at your example from Mission Impossible. We are so used to impractical stunts in action films that we rarely think twice about them. It's fun to pick them apart a little and see how they stand up to the real world. Aside from rolling our eyes at the most egregious nonsense (a fridge protects you from a nuclear explosion, anyone?) I think most of us simply don't question them. Suspension of disbelief is needed to accept all kinds of movie scenes - except for those in which our ignorance of the principles involved accomplishes the same thing!
But I think that we probably notice more when we are already being critical of the film for one reason or another, or when there are a string of apparently improbable events in a row. I think that if one does not like other choices made in the Goblintown sequence, whether they are faults of writing, design, or what have you, one is probably more likely to think the bridge scene absurd, while, as you demonstrate, it may indeed be possible. For instance, someone who was annoyed at the Stone Giants sequence which immediately preceded the Goblintown section of the film, and then was further irritated by Bilbo trying to leave and then the change from having the dwarves fall down a trap door and have a water-slide like fall into Goblintown instead of being captured by goblins leaping out of a secret door and herding them down, might well take the bridge fall as "the final straw" rather than being able to judge it on its own separate merits. Perhaps some of the anguish over this sequence is simply accumulated frustration from other changes not liked, so that taking an objective view of it becomes much more difficult. I wonder if there is a formula to measure the exponential increase of dislike created by having a string of altered scenes in a row as opposed to that created by an equal number of story changes more evenly distributed between canonical scenes?
Speaking of which, I know the perspective makes it difficult to see how steep the tunnel was, but how did the initial fall down the trap door chute strike you? Was that more or less possible to survive without a scratch than the bridge fall?
After reading this, I can't wait to read your barrel analysis in Part 3 at the next Symposium!
Silverlode
"Dark is the water of Kheled-zâram, and cold are the springs of Kibil-nâla, and fair were the many-pillared halls of Khazad-dűm in Elder Days before the fall of mighty kings beneath the stone."
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Aunt Dora Baggins
Immortal
Jul 22 2013, 3:24am
Post #25 of 68
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But when I click on the link I just get a blank page :-(
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"For DORA BAGGINS in memory of a LONG correspondence, with love from Bilbo; on a large wastebasket. Dora was Drogo's sister, and the eldest surviving female relative of Bilbo and Frodo; she was ninety-nine, and had written reams of good advice for more than half a century."
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"A Chance Meeting at Rivendell" and other stories
leleni at hotmail dot com
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