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**Jan_**
Will have to go over link later, thanx!

Stepson became engineer(and Master's last yr.!), but i'm no school boy; just (well researched) treeman slop of made up imagery to observations and research etc. All the alpha,theta,sigma symbols seen in this stuff kinda look like Greek to me..

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The 120degree spread line equal to each side as 60 degree angle deflection and equal force on each side of redirect pulley system (ignoring frictions).

will give co(lumn)sine= .5 and sin(e)=.86 that i use as percentages of support (column) to purpose against load and sin of wrenching sideways forces against respectively of support force (or length)Thus, the 2 legs of line will place 2xLine tension x .5cosine = just original line tension on the support YES same as that angle spread would hobble a 2/1 system to only give 1x return! Closing angle more will increase this beyond line tension YES as is closer to a 2/1 of closed angle/ Zer0 deflection to get all the 2x potential untaxed by any side force here or in 2/1.

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BUT, for a 60 degree line (1 side leg of 120 spread), to place 100# upward support on climber increases line tension :

100#needed ÷ .5co(lumn)sine of efficiency of 60 degrees = 200# line tension.

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So that 200# line tension x .5co(lumn)sine = 100# upward support from 200# line tension cash

BUT,

that means 200#line tension x .86sin(e) = 172# pulling at right angle on climber to get the 100# upward support!!

(Short cut need 100# @60 degrees side force =100# support column needed x 1.72(tangent of 60)= 172# sideforce.)

Must always look to sides both ways when crossing any line of force on this street and at what could be coming and it's costs to not get run over!!

Similarly if had 200in. 2x4 leaned @ 60 degrees,

Rising support column reaches 200 of available x .5 co(lumn)sine would give 100in. rise,

But throw to side 200 available x .86 sin = 172in. in doing so. Same Math to see if length quantity is enough as for then to see if once reached enough force etc.

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For me,

Co(lumn)sine angle directly opposing load direction unique single dimension column of support.

Sin(e) is any thing to the other 2 dimensions, side forces not on the benchmark reference column of loading. As co(lumn)sine is the support, sin(e) is it's non. Sine is tax extruded at that angle to give support, thus from the 'wallet' of available line tension at the moment, some of the money goes to sin(e), so less goes to actual support of co(lumn)sine to work done. Need more support, need more line tension(cash) and/or less tax to 'net' the increased support.

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i also say cosine is to your cos(cause), vs. the sin against. Some politicians are always letting their sin over run their cos!

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Temple roof stood on pure vertical support co(lumn)sines, Samson came across on sin angle; needing equal opposite of other column to do so. Cuz 90degree from support is Zer0 support on that cross axis!

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Glass, granite, marble slabs is a fave model of carrying by numbers to protect the minimal axis of least support from overloading; typically carried vertical +10.

>>If pure vertical could throw side force either way to more impact, definitive lean gives 1 angle predictable lean and then support backing.

>>typically at ~10 degrees; giving cos: .984 sin:.173

translates to 98.4% of weight borne down column properly;

length handles forward/backward forces thru the longest column.

Minimal side forces are of weight x 17.3% already @ 10 degrees!!!

note how cosine creeps down slow to start, as sine throws to side HARD!

(reverse effect after 60 degrees, cosine falls fast, sine creeps)

So minimal/thinnest axis gets minimal force, and then supported.

Greater forces, are carried properly on the more major axises of length and height, rather than thinness. Some tile shipped pure vertical packages etc.

Thus you don't lay weighty slabs down or in direction of travel usually, to protect the minimal support axis of thinness, thereby weakness.

This is how these delicates are ushered thru the forces; as an extreme comparative force model for what we face.

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These principles are so all around us, hard to see and sift. Even realistic game shadow gradient fades are on cosine/sine scales or look fakey, our protecting animal eye knows when is not what it sees all the time!

Above are some of the imageries, reminders, models, mnemonics have carried and digested to keep self inline and weigh and measure what i see/feel.

Hope this helps!

Spaghetti theory : Gotta keep throwing at self until sticks!

This stuff is pure gold once get it, so many applications!