en>Jburlinson: /* Semantics of modal logic */ add cites

2014-12-24T22:07:03Z

Semantics of modal logic: add cites

← Older revision		Revision as of 00:07, 25 December 2014
Line 1:		Line 1:
	[[File:Fall factor diagram.svg\|thumb\|250px\|The climber will fall about the same height ''h'' in both cases, but she will be subjected to a greater force at position 1, due to the greater fall factor.]]
	In [[climbing]] (specifically in [[lead climbing]]) using a [[dynamic rope]], the '''fall factor''' ('''''f''''') is the ratio of the height (''h'') a climber falls before the climber's rope begins to stretch and the rope length (''L'') available to absorb the energy of the fall.

	~~:<math>f = \frac{h}{L}</math>~~

	~~== Impact force ==~~		I am Milford and I totally love this title. Playing mah jongg is a thing that she is completely addicted to. I've usually [http://www.britannica.com/search?query=cherished+residing cherished residing] in Hawaii and my mothers and fathers reside close by. I am currently a production and distribution officer. Go to my [http://browse.deviantart.com/?qh=&section=&global=1&q=web+site web site] to discover out much more: http://delhipower.in/top-guide-of-nya-online-casino/<br><br>My weblog: [http://delhipower.in/top-guide-of-nya-online-casino/ nya online casinon på nätet]

	The impact force is defined as the maximum tension in the rope when a climber falls. Using the common rope model of an undamped [[harmonic oscillator]] (HO) the impact force ''F<sub>max</sub>'' in the rope is given by:

	~~:<math>F_{max} = mg + \sqrt{(mg)^2 + 2mghk} = mg + \sqrt{(mg)^2 + 2mgEqf}</math>~~

	~~where ''mg'' is the climber's weight, ''h'' is the fall height~~ and ~~''k'' is the spring constant of the rope~~.~~{{Citation needed\|date=October 2013}} Using the [[elasticity modulus]] ''E'' = ''k L/q'' which~~ is a material constant, the impact force depends only on the fall factor ''f'', i.e. on the ratio ''h/L'', the cross section ''q'' of the rope and the climber’s weight. The more rope is available, the softer the rope becomes which is just compensating the higher fall energy. The maximum force on the climber is ''F<sub>max</sub>'' reduced by the climber’s weight ''mg''. The above formula can be easily obtained by the law of conservation of energy at the time of maximum tension resp. maximum elongation ''x<sub>max</sub>'' of the rope:

	~~:<math> mgh = \frac{1}{2}kx_{max}^2 - mgx_{max}\ ; \ F_{max} = k x_{max} </math>~~

	~~Using the HO model~~ to ~~obtain the impact force of real climbing ropes as a function of fall height ''h'' and climber's weight ''mg'', one must know the experimental value for ''E'' of a given rope~~. ~~However, rope manufacturers give only the rope’s impact force~~ '~~'F<sub>0</sub>'' and its static and dynamic elongations that are measured under standard~~ [[UIAA]] fall conditions: A fall height ''h<sub>0</sub>'' of 2 x 2.3 m with an available rope length ''L<sub>0</sub>'' = 2.6m leads to a fall factor ''f<sub>0</sub>'' = ''h<sub>0</sub>/L<sub>0</sub>'' = 1.77 and a fall velocity ''v<sub>0</sub>'' = (''2gh<sub>0</sub>'')<sup>1/2</sup> = 9.5 m/s at the end of falling the distance ''h<sub>0</sub>''. The mass ''m<sub>0</sub>'' used in the fall is 80 kg. Using these values to eliminate the unknown quantity ''E'' leads to an expression of the impact force as a function of arbitrary fall heights ''h'' and arbitrary fall factors ''f'' of the form:<ref name=leuthaeusser>Ulrich Leuthäusser (2010):{{cite web\|url=http://www.~~sigmadewe~~.com/~~klettersport-themen.html~~?~~&L=1\|title=Viscoelastic theory of climbing ropes\|accessdate =2010-12-02}}Retrieved December 2, 2010</ref>~~

	~~:<math>F_{max}~~ = mg + ~~\sqrt{(mg)^2 + F_0(F_0-2m_0g)\frac{m}{m_0}\frac{f}{f_0}} </math>~~

	~~[[Image:Physics of climbing ropes.png\|thumb\|Impact force as a function of dynamic elongation^(-1) for different friction constants κ.~~]] This simple undamped harmonic oscillator model of a rope, however, cannot explain real ropes. First, it is evident that real ropes hardly oscillate after a fall. After one period the rope has settled and ~~stopped oscillating. The HO also cannot explain correctly the experimental values of a climbing rope such as its static~~ and ~~dynamic elongation and the correct relations to its impact force. This can be corrected only~~ by ~~considering friction in the rope~~. ~~On the basis of~~ a Viscoelastic Standard Linear Solid model one gets more complicated expressions for impact force and static and dynamic elongations.<ref name=leuthaeusser /> Friction in the rope leads to energy dissipation and ~~thus to a reduction of the impact force compared to the undamped harmonic oscillator model~~. ~~It also leads~~ to an additional elongation of the rope. The diagram shows how the impact forces of real climbing ropes under standard UIAA fall conditions relate to their measured dynamic elongations. It also shows that the HO model cannot explain these dependencies of real climbing ropes.

	~~When the rope is clipped into several carabiners between the climber and the~~ [[belayer]], an additional type of friction occurs, the so-called dry [[friction]] between the rope and particularly the last clipped carabiner. Dry friction leads to an effective rope length smaller than the available length ''L'' and thus increases the impact force.<ref name=uleuthaeusser>Ulrich Leuthäusser (2011):{{cite web\|url=http://~~www~~.~~sigmadewe~~.com/~~fileadmin/user_upload/pdf-Dateien/Physics_of_climbing_ropes_Part_2.pdf~~?&L=1~~\|title~~=~~Physics of climbing ropes: impact forces, fall factors and rope drag\|accessdate =2011-01-15}}Retrieved January 15, 2011</ref> Dry friction is also responsible for the rope drag a climber has~~ to overcome in order to move forward. It can be expressed by an effective mass of the rope that the climber has to pull which is always larger than the rope mass itself. It depends exponentially on the sum of the angles of the direction changes the climber has made.<ref name=uleuthaeusser />

	~~== Lead climbing ==~~

	A fall factor of two is the maximum that is possible in a [[lead climbing]] fall, since the length of an arrested fall cannot exceed two times the length of the rope. Normally, a factor-2 fall can occur only when a lead climber who has placed no [[Protection (climbing)\|protection]] falls past the belayer (two times the distance of the rope length between them), or the anchor if the climber is solo climbing the route using a self-belay. As soon as the climber clips the rope into protection above the belay, the distance of the potential fall as a function of rope length is lessened, and the fall factor drops below 2.

	~~A fall of 20 feet is~~ much more severe (exerts more force on the climber and climbing equipment) if it occurs with 10 feet of rope out (i.e. the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall) than if it occurs 100 feet above the belayer (a fall factor of 0.2), in which case the stretch of the rope more effectively cushions the fall.

	~~== Fall factors above two ==~~

	In falls occurring on a [[via ferrata]], fall factors can be much higher. This is possible because the length of rope between harness and [[carabiner]] is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable.

	~~==See also==~~
	*[[Whipper]]

	~~== References ==~~
	~~<references/>~~

	~~== External links ==~~
	*{{cite web
	~~\| last = Goldstone~~
	~~\| first = Richard~~
	~~\| coauthors =~~
	~~\| title = The Standard Equation for Impact Force~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date = December 27, 2006~~
	~~\| url =~~ http://~~www~~.~~rockclimbing.com~~/~~cgi~~-~~bin/forum/gforum.cgi?do=post_attachment;postatt_id=746~~
	~~\| doi =~~
	~~\| accessdate = 2009~~-04-~~17}}~~

	*{{cite web
	~~\| last = Busch~~
	~~\| first = Wayne~~
	~~\| coauthors =~~
	~~\| title = Climbing Physics~~ - ~~Understanding Fall Factors~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date =~~
	~~\| url =~~ http://~~www~~.~~southeastclimbing.com~~/~~faq/faq_fall_factor.htm~~
	~~\| doi =~~
	~~\| accessdate = 2008~~-06-~~14}}~~

	*{{cite web
	~~\| title = UKC~~ - ~~Understanding fall factors~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date =~~
	~~\| url = http:~~/~~/www.ukclimbing.com/articles/page.php?id=647~~
	~~\| format =~~
	~~\| doi = }}~~

	~~{{DEFAULTSORT:Fall Factor}}~~
	~~[[Category:Climbing]]~~
	~~[[Category:Mechanics]~~]

en>AugPi: /* Semantics of intuitionistic logic */ (+)

2014-01-28T16:00:56Z

Semantics of intuitionistic logic: (+)

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en>EmilJ: Undid revision 501084963 by Linas (talk) MOS:ABBR#Latin abbreviations

2012-07-07T19:19:02Z

Undid revision 501084963 by Linas (talk) MOS:ABBR#Latin abbreviations

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← Older revision		Revision as of 00:07, 25 December 2014
Line 1:		Line 1:
	[[File:Fall factor diagram.svg\|thumb\|250px\|The climber will fall about the same height ''h'' in both cases, but she will be subjected to a greater force at position 1, due to the greater fall factor.]]
	In [[climbing]] (specifically in [[lead climbing]]) using a [[dynamic rope]], the '''fall factor''' ('''''f''''') is the ratio of the height (''h'') a climber falls before the climber's rope begins to stretch and the rope length (''L'') available to absorb the energy of the fall.

	~~:<math>f = \frac{h}{L}</math>~~

	~~== Impact force ==~~		I am Milford and I totally love this title. Playing mah jongg is a thing that she is completely addicted to. I've usually [http://www.britannica.com/search?query=cherished+residing cherished residing] in Hawaii and my mothers and fathers reside close by. I am currently a production and distribution officer. Go to my [http://browse.deviantart.com/?qh=&section=&global=1&q=web+site web site] to discover out much more: http://delhipower.in/top-guide-of-nya-online-casino/<br><br>My weblog: [http://delhipower.in/top-guide-of-nya-online-casino/ nya online casinon på nätet]

	The impact force is defined as the maximum tension in the rope when a climber falls. Using the common rope model of an undamped [[harmonic oscillator]] (HO) the impact force ''F<sub>max</sub>'' in the rope is given by:

	~~:<math>F_{max} = mg + \sqrt{(mg)^2 + 2mghk} = mg + \sqrt{(mg)^2 + 2mgEqf}</math>~~

	~~where ''mg'' is the climber's weight, ''h'' is the fall height~~ and ~~''k'' is the spring constant of the rope~~.~~{{Citation needed\|date=October 2013}} Using the [[elasticity modulus]] ''E'' = ''k L/q'' which~~ is a material constant, the impact force depends only on the fall factor ''f'', i.e. on the ratio ''h/L'', the cross section ''q'' of the rope and the climber’s weight. The more rope is available, the softer the rope becomes which is just compensating the higher fall energy. The maximum force on the climber is ''F<sub>max</sub>'' reduced by the climber’s weight ''mg''. The above formula can be easily obtained by the law of conservation of energy at the time of maximum tension resp. maximum elongation ''x<sub>max</sub>'' of the rope:

	~~:<math> mgh = \frac{1}{2}kx_{max}^2 - mgx_{max}\ ; \ F_{max} = k x_{max} </math>~~

	~~Using the HO model~~ to ~~obtain the impact force of real climbing ropes as a function of fall height ''h'' and climber's weight ''mg'', one must know the experimental value for ''E'' of a given rope~~. ~~However, rope manufacturers give only the rope’s impact force~~ '~~'F<sub>0</sub>'' and its static and dynamic elongations that are measured under standard~~ [[UIAA]] fall conditions: A fall height ''h<sub>0</sub>'' of 2 x 2.3 m with an available rope length ''L<sub>0</sub>'' = 2.6m leads to a fall factor ''f<sub>0</sub>'' = ''h<sub>0</sub>/L<sub>0</sub>'' = 1.77 and a fall velocity ''v<sub>0</sub>'' = (''2gh<sub>0</sub>'')<sup>1/2</sup> = 9.5 m/s at the end of falling the distance ''h<sub>0</sub>''. The mass ''m<sub>0</sub>'' used in the fall is 80 kg. Using these values to eliminate the unknown quantity ''E'' leads to an expression of the impact force as a function of arbitrary fall heights ''h'' and arbitrary fall factors ''f'' of the form:<ref name=leuthaeusser>Ulrich Leuthäusser (2010):{{cite web\|url=http://www.~~sigmadewe~~.com/~~klettersport-themen.html~~?~~&L=1\|title=Viscoelastic theory of climbing ropes\|accessdate =2010-12-02}}Retrieved December 2, 2010</ref>~~

	~~:<math>F_{max}~~ = mg + ~~\sqrt{(mg)^2 + F_0(F_0-2m_0g)\frac{m}{m_0}\frac{f}{f_0}} </math>~~

	~~[[Image:Physics of climbing ropes.png\|thumb\|Impact force as a function of dynamic elongation^(-1) for different friction constants κ.~~]] This simple undamped harmonic oscillator model of a rope, however, cannot explain real ropes. First, it is evident that real ropes hardly oscillate after a fall. After one period the rope has settled and ~~stopped oscillating. The HO also cannot explain correctly the experimental values of a climbing rope such as its static~~ and ~~dynamic elongation and the correct relations to its impact force. This can be corrected only~~ by ~~considering friction in the rope~~. ~~On the basis of~~ a Viscoelastic Standard Linear Solid model one gets more complicated expressions for impact force and static and dynamic elongations.<ref name=leuthaeusser /> Friction in the rope leads to energy dissipation and ~~thus to a reduction of the impact force compared to the undamped harmonic oscillator model~~. ~~It also leads~~ to an additional elongation of the rope. The diagram shows how the impact forces of real climbing ropes under standard UIAA fall conditions relate to their measured dynamic elongations. It also shows that the HO model cannot explain these dependencies of real climbing ropes.

	~~When the rope is clipped into several carabiners between the climber and the~~ [[belayer]], an additional type of friction occurs, the so-called dry [[friction]] between the rope and particularly the last clipped carabiner. Dry friction leads to an effective rope length smaller than the available length ''L'' and thus increases the impact force.<ref name=uleuthaeusser>Ulrich Leuthäusser (2011):{{cite web\|url=http://~~www~~.~~sigmadewe~~.com/~~fileadmin/user_upload/pdf-Dateien/Physics_of_climbing_ropes_Part_2.pdf~~?&L=1~~\|title~~=~~Physics of climbing ropes: impact forces, fall factors and rope drag\|accessdate =2011-01-15}}Retrieved January 15, 2011</ref> Dry friction is also responsible for the rope drag a climber has~~ to overcome in order to move forward. It can be expressed by an effective mass of the rope that the climber has to pull which is always larger than the rope mass itself. It depends exponentially on the sum of the angles of the direction changes the climber has made.<ref name=uleuthaeusser />

	~~== Lead climbing ==~~

	A fall factor of two is the maximum that is possible in a [[lead climbing]] fall, since the length of an arrested fall cannot exceed two times the length of the rope. Normally, a factor-2 fall can occur only when a lead climber who has placed no [[Protection (climbing)\|protection]] falls past the belayer (two times the distance of the rope length between them), or the anchor if the climber is solo climbing the route using a self-belay. As soon as the climber clips the rope into protection above the belay, the distance of the potential fall as a function of rope length is lessened, and the fall factor drops below 2.

	~~A fall of 20 feet is~~ much more severe (exerts more force on the climber and climbing equipment) if it occurs with 10 feet of rope out (i.e. the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall) than if it occurs 100 feet above the belayer (a fall factor of 0.2), in which case the stretch of the rope more effectively cushions the fall.

	~~== Fall factors above two ==~~

	In falls occurring on a [[via ferrata]], fall factors can be much higher. This is possible because the length of rope between harness and [[carabiner]] is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable.

	~~==See also==~~
	*[[Whipper]]

	~~== References ==~~
	~~<references/>~~

	~~== External links ==~~
	*{{cite web
	~~\| last = Goldstone~~
	~~\| first = Richard~~
	~~\| coauthors =~~
	~~\| title = The Standard Equation for Impact Force~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date = December 27, 2006~~
	~~\| url =~~ http://~~www~~.~~rockclimbing.com~~/~~cgi~~-~~bin/forum/gforum.cgi?do=post_attachment;postatt_id=746~~
	~~\| doi =~~
	~~\| accessdate = 2009~~-04-~~17}}~~

	*{{cite web
	~~\| last = Busch~~
	~~\| first = Wayne~~
	~~\| coauthors =~~
	~~\| title = Climbing Physics~~ - ~~Understanding Fall Factors~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date =~~
	~~\| url =~~ http://~~www~~.~~southeastclimbing.com~~/~~faq/faq_fall_factor.htm~~
	~~\| doi =~~
	~~\| accessdate = 2008~~-06-~~14}}~~

	*{{cite web
	~~\| title = UKC~~ - ~~Understanding fall factors~~
	~~\| work =~~
	~~\| publisher =~~
	~~\| date =~~
	~~\| url = http:~~/~~/www.ukclimbing.com/articles/page.php?id=647~~
	~~\| format =~~
	~~\| doi = }}~~

	~~{{DEFAULTSORT:Fall Factor}}~~
	~~[[Category:Climbing]]~~
	~~[[Category:Mechanics]~~]

Kripke semantics - Revision history

en>Jburlinson: /* Semantics of modal logic */ add cites

en>AugPi: /* Semantics of intuitionistic logic */ (+)

en>EmilJ: Undid revision 501084963 by Linas (talk) MOS:ABBR#Latin abbreviations