9010catenary.fodt

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       AAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">From geometry</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object3" text:anchor-type="as-char" svg:width="0.7661in" svg:height="0.4043in" draw:z-index="2"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>cos</mi>
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          <mfrac>
           <mi mathvariant="italic">dx</mi>
           <mi mathvariant="italic">ds</mi>
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         </mrow>
        </mrow>
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       </semantics>
      </math>
     </draw:object><draw:image>
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       AAAAAACWAAEAAgAAAAkAjAABAAAAAAA=
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object4" text:anchor-type="as-char" svg:width="0.7429in" svg:height="0.4043in" draw:z-index="3"><draw:object>
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       <semantics>
        <mrow>
         <mi>sin</mi>
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        <annotation encoding="StarMath 5.0">sin %theta = dy over ds</annotation>
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     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">For variation</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object5" text:anchor-type="as-char" svg:width="1.9008in" svg:height="0.4516in" draw:z-index="4"><draw:object>
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       <semantics>
        <mrow>
         <mi>cos</mi>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mrow>
             <mrow>
              <mi mathvariant="normal">θ</mi>
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              <mi>d</mi>
             </mrow>
             <mi mathvariant="normal">θ</mi>
            </mrow>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mrow>
           <mfrac>
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            <mi mathvariant="italic">ds</mi>
           </mfrac>
           <mo stretchy="false">+</mo>
           <mfrac>
            <mrow>
             <msup>
              <mi>d</mi>
              <mn>2</mn>
             </msup>
             <mi>x</mi>
            </mrow>
            <msup>
             <mi mathvariant="italic">ds</mi>
             <mn>2</mn>
            </msup>
           </mfrac>
          </mrow>
         </mrow>
         <mi mathvariant="italic">ds</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">cos left(%theta + d%theta right) = dx over ds 
+ {d^2 x} over {ds^2}
ds</annotation>
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     </draw:image>
    </draw:frame></text:p>
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       <semantics>
        <mrow>
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             <mrow>
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          </mrow>
          <mo stretchy="false">=</mo>
          <mrow>
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             <mi>y</mi>
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            <msup>
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           </mfrac>
          </mrow>
         </mrow>
         <mi mathvariant="italic">ds</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">sin left(%theta + d%theta right) = dy over ds 
+ {d^2 y} over {ds^2}
ds</annotation>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Substitution into the force equations</text:p>
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-
T
dx over ds 
=
0
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T + dT
right)
left(
dy over ds 
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ds
right)
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T
dy over ds 
-
%rho g ds
=
0
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   <text:p text:style-name="Standard">Simplifying and neglecting higher order terms</text:p>
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+
dT over ds
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=
0
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-
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=
0
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   <text:p text:style-name="Standard">Using the identity</text:p>
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+
left(dy over ds right)^2
=
1
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   <text:p text:style-name="Standard">Differentiation wrt ds</text:p>
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dx over ds 
{d^2 x} over {ds^2}
+
2
dy over ds 
{d^2 y} over {ds^2}
=
0
</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard">Summing the two force equations and using the identity relations above</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object18" text:anchor-type="as-char" svg:width="1.1689in" svg:height="0.4043in" draw:z-index="17"><draw:object>
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         <mi>g</mi>
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           <mi mathvariant="italic">ds</mi>
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          <mo stretchy="false">=</mo>
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         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">dT over ds
-
%rho g
dy over ds
=
0
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:p text:style-name="Standard">Dividing by dx/ds</text:p>
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       <semantics>
        <mrow>
         <mrow>
          <mfrac>
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          </mfrac>
          <mo stretchy="false">−</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <mfrac>
           <mi mathvariant="italic">dy</mi>
           <mi mathvariant="italic">dx</mi>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">dT over dx
-
%rho g
dy over dx
=
0
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYA
       AQAEAAAAAAAAAGcAAQAQAAAAgwYAAPcBAABbCAAACwIAAIwAAQAAAAAAiwABAAIAAAAfAIoA
       AQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAgD/AwAAAAAA
       AAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAA
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       AAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAAB6CAAAewIA
       AAEAAAA9AFMBAAAAAP//AQA9AIwAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBU
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       AgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA0gkAAHsCAAABADDUAAAA
       AAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Dividing by dy/ds</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object19" text:anchor-type="as-char" svg:width="0.9457in" svg:height="0.4043in" draw:z-index="39"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mfrac>
           <mi mathvariant="italic">dT</mi>
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          </mfrac>
          <mo stretchy="false">−</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mrow>
          <mi>g</mi>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">dT over dy
-
%rho g
=
0
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxlAAAAAAAApwEAAAAAAAAAAAUAAAAA
       AAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAA
       AAByAAIAFwAAAEUFAAB7AgAAAQBn1AAAAAAA//8BAGcAjAABAAAAAACLAAEAAgAAAB8AigAB
       ADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACnAQAA//8AAAAABQAAAAAAAAD/AwAAAAAA
       AAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAA
       TgYAAHsCAAABAAAAPQBTAQAAAAD//wEAPQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAOwAAAAMA
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       AAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAKYHAAB7AgAA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Integrating wrt to y</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object20" text:anchor-type="as-char" svg:width="1.1335in" svg:height="0.2299in" draw:z-index="18"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mi>T</mi>
          <mo stretchy="false">=</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
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            <mo stretchy="false">+</mo>
            <mi>C</mi>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">T 
=
%rho g 
left(
y + C
right)
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AIYAAQAEAAAAAAAAAHIAAgAaAAAAjQEAAI0BAAABAAAAPQBTAQAAAAD//wEAPQCMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAF
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       AAAAAAAAcgACABoAAADlAgAAjQEAAAEAAADBA+UAAAAAAP//AQDBA4wAAQAAAAAAiwABAAIA
       AAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAgD/
       AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA
       AgAXAAAAIgQAAI0BAAABAGfUAAAAAAD//wEAZwCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAA
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       AAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA5AUAAI0BAAABAHm6
       AAAAAAD//wEAeQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv
       bAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH
       AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAAC4BgAAjQEAAAEAAAArAFMBAAAAAP//
       AQArAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACn
       AQAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/
       ////AIYAAQAEAAAAAAAAAHIAAgAXAAAAEAgAAI0BAAABAEMXAQAAAAD//wEAQwCMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAA2AAAAOwBAAD//wAAAAAF
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Substituting T and dT/ds into the second force equation</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object21" text:anchor-type="as-char" svg:width="2.722in" svg:height="0.4516in" draw:z-index="19"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">ρ</mi>
         <mi>g</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
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            <mo stretchy="false">+</mo>
            <mi>C</mi>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
         <mrow>
          <mfrac>
           <mrow>
            <msup>
             <mi>d</mi>
             <mn>2</mn>
            </msup>
            <mi>y</mi>
           </mrow>
           <msup>
            <mi mathvariant="italic">ds</mi>
            <mn>2</mn>
           </msup>
          </mfrac>
          <mo stretchy="false">+</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <msup>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">(</mo>
            <mrow>
             <mfrac>
              <mi mathvariant="italic">dy</mi>
              <mi mathvariant="italic">ds</mi>
             </mfrac>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">)</mo>
           </mrow>
           <mn>2</mn>
          </msup>
          <mo stretchy="false">−</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mrow>
          <mi>g</mi>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%rho g 
left(
y + C
right)

{d^2 y} over {ds^2}
+
%rho g
left( 
dy over ds
right)^2
-
%rho g
=
0
</annotation>
       </semantics>
      </math>
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       AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAADtBgAA5QIAAAEAAAApAGYAAAAAAP//
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left
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=
s^2 
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+ 2 bar C_3
=
left(
s
+ C_2
right)^2 
- C^2_2
+ 2 bar C_3
=
left(
s
+ C_2
right)^2 
+ C^2_3</annotation>
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   <text:p text:style-name="Standard">Substituting T and dT/ds into the first force equation</text:p>
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          <mi mathvariant="italic">ds</mi>
         </mfrac>
         <mrow>
          <mfrac>
           <mi mathvariant="italic">dx</mi>
           <mi mathvariant="italic">ds</mi>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%rho g left( y + C right)
{d^2 x} over {ds^2}
+
%rho g
dy over ds
dx over ds
=
0
</annotation>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Substituting <draw:frame draw:style-name="fr2" draw:name="Object30" text:anchor-type="as-char" svg:width="0.6181in" svg:height="0.2299in" draw:z-index="30"><draw:object>
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       <semantics>
        <mrow>
         <mo fence="true" form="prefix" stretchy="true">(</mo>
         <mrow>
          <mrow>
           <mi>y</mi>
           <mo stretchy="false">+</mo>
           <mi>C</mi>
          </mrow>
         </mrow>
         <mo fence="true" form="postfix" stretchy="true">)</mo>
        </mrow>
        <annotation encoding="StarMath 5.0">left(y + C right)</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AQAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/
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      </office:binary-data>
     </draw:image>
    </draw:frame>from the y equation</text:p>
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               </msub>
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             </mrow>
             <mo fence="true" form="postfix" stretchy="true">)</mo>
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            <mn>2</mn>
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            <mn>3</mn>
            <mn>2</mn>
           </msubsup>
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         </msqrt>
         <mrow>
          <mfrac>
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           <mi mathvariant="italic">ds</mi>
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          <mo stretchy="false">=</mo>
          <msub>
           <mi>C</mi>
           <mn>4</mn>
          </msub>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">sqrt
{ 
left(
s
+ C_2
right)^2 
+ C^2_3
}
{dx} over {ds}
=
C_4
</annotation>
       </semantics>
      </math>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object32" text:anchor-type="as-char" svg:width="1.4965in" svg:height="0.5161in" draw:z-index="32"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="italic">dx</mi>
         <mo stretchy="false">=</mo>
         <mfrac>
          <msub>
           <mi>C</mi>
           <mrow>
            <mn>4</mn>
            <mi mathvariant="italic">ds</mi>
           </mrow>
          </msub>
          <msqrt>
           <mrow>
            <msup>
             <mrow>
              <mo fence="true" form="prefix" stretchy="true">(</mo>
              <mrow>
               <mrow>
                <mi>s</mi>
                <mo stretchy="false">+</mo>
                <msub>
                 <mi>C</mi>
                 <mn>2</mn>
                </msub>
               </mrow>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
             <mn>2</mn>
            </msup>
            <mo stretchy="false">+</mo>
            <msubsup>
             <mi>C</mi>
             <mn>3</mn>
             <mn>2</mn>
            </msubsup>
           </mrow>
          </msqrt>
         </mfrac>
        </mrow>
        <annotation encoding="StarMath 5.0">dx
=
{
C_4
ds}
over {
sqrt{ 
left(
s
+ C_2
right)^2 
+ C^2_3}}
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Integrating</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr2" draw:name="Object33" text:anchor-type="as-char" svg:width="1.9744in" svg:height="0.4992in" draw:z-index="33"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mrow>
           <mi>x</mi>
           <mo stretchy="false">+</mo>
           <msub>
            <mi>C</mi>
            <mn>5</mn>
           </msub>
          </mrow>
          <mo stretchy="false">=</mo>
          <msub>
           <mi>C</mi>
           <mn>4</mn>
          </msub>
         </mrow>
         <msup>
          <mi>sinh</mi>
          <mrow>
           <mo stretchy="false">−</mo>
           <mn>1</mn>
          </mrow>
         </msup>
         <mrow>
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=
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sinh^-1
left(
{
s
+ C_2}
over 
{C_3}
right)

</annotation>
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left(
{
x
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over 
{C_4}
right)
=
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{C_3}
=
{
sqrt
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over 
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        <annotation encoding="StarMath 5.0">s=
C_3
sinh
left(
{
x
+ C_5}
over 
{C_4}
right)
-
C_2
</annotation>
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C_3
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1
+
sinh^2
left(
{
x
+ C_5}
over 
{C_4}
right)
}
-
C

newline
&quot;&quot;~~
=
C_3
cosh
left(
{
x
+ C_5}
over 
{C_4}
right)
-
C
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over
dx
=
C_3
sinh
left(
{
x
+ C_5}
over 
{C_4}
right)
</annotation>
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           <mrow>
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        <annotation encoding="StarMath 5.0">dy
over
ds
(x)
=
{
sinh
left(
{
x
+ C_5}
over 
{C_4}
right)
}
over
{
sqrt
{
C^2_3
+
sinh^2
left(
{
x
+ C_5}
over 
{C_4}
right)

}
}
</annotation>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Anchors</text:p>
   <text:p text:style-name="Standard">There are two portions to an anchor line: a portion lying on the ground and a catenary portion. The total length is fixed for a given length. The anchor line is straight in the plan view.</text:p>
   <text:p text:style-name="Standard">Let x be the horizontal coordinate. x1 be the flat portion. x2 be the given x coordinate of the moving end of the line.</text:p>
   <text:p text:style-name="Standard">Let y be the vertical coordinate from the ground on up. y2 be the given y coordinate of the moving end of the line.</text:p>
   <text:p text:style-name="Standard">When the anchor line is slack, the unknowns are:</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>C</mi>
         <mi>,</mi>
         <msub>
          <mi>C</mi>
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         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>2</mn>
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         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>3</mn>
         </msub>
         <mi>,</mi>
         <msub>
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         </msub>
         <mi>,</mi>
         <msub>
          <mi>x</mi>
          <mn>1</mn>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">C,C_1,C_2,C_3,C_4,x_1</annotation>
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   <text:p text:style-name="Standard">The boundary conditions are:</text:p>
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     </draw:image>
    </draw:frame></text:p>
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        <annotation encoding="StarMath 5.0">s left(x_2 right) = l</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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    </draw:frame></text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object38" text:anchor-type="as-char" svg:width="0.7374in" svg:height="0.2335in" draw:z-index="36"><draw:object>
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        </mrow>
        <annotation encoding="StarMath 5.0">y left(x_1 right) = 0</annotation>
       </semantics>
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          <mrow>
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          <mo stretchy="false">=</mo>
          <mi>h</mi>
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        <annotation encoding="StarMath 5.0">y left(x_2 right) = h</annotation>
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       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHoFAACNAQAAAQBo1AAAAAAA//8BAGgA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object39" text:anchor-type="as-char" svg:width="0.8335in" svg:height="0.4043in" draw:z-index="58"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mfrac>
          <mi mathvariant="italic">dy</mi>
          <mi mathvariant="italic">dx</mi>
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         <mrow>
          <mrow>
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          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">dy over dx left(x_1 right) 
= 
0</annotation>
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       AAAAAAAApwEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAJ8AAAByAQAAAgBkeY4BAAAAAP//AgBk
       AHkAjAABAAAAAACLAAEAAgAAAB8AhQABAAUAAAAAAAAAAYQAAQAFAAAAAAAAAACKAAEAPAAA
       AAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA
       /wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAZwABABAAAACEAAAA
       9wEAAFwCAAALAgAAjAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFs
       ZQAAAAAAAKcBAAAAAAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH
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       /////wCGAAEABAAAAAAAAAByAAIAGgAAAJUCAACwAgAAAQAAACgAdwAAAAAA//8BACgAjAAB
       AAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAKcBAAAAAAAA
       AAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgAB
       AAQAAAAAAAAAcgACABcAAAA0AwAAewIAAAEAeLoAAAAAAP//AQB4AIwAAQAAAAAAiwABAAIA
       AAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAABQAAAAAAAAD/
       AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA
       AgAXAAAACAQAAOUCAAABADF/AAAAAAD//wEAMQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAA
       AAMANgAAAAoAT3BlblN5bWJvbAAA9AAAAC0CAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA
       /wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAACMBAAA
       sAIAAAEAAAApAHcAAAAAAP//AQApAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAA
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       iAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAABAFAAB7AgAAAQAA
       AD0AUwEAAAAA//8BAD0AjAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3Ju
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       AQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABoBgAAewIAAAEAMNQAAAAAAP//
       AQAwAIwAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Summing forces in y for the catenary portion:</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object42" text:anchor-type="as-char" svg:width="1.8717in" svg:height="0.4043in" draw:z-index="40"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>T</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="false">(</mo>
          <mrow>
           <mi>h</mi>
          </mrow>
          <mo fence="true" form="postfix" stretchy="false">)</mo>
         </mrow>
         <mfrac>
          <mi mathvariant="italic">dy</mi>
          <mi mathvariant="italic">ds</mi>
         </mfrac>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <msub>
             <mi>x</mi>
             <mn>2</mn>
            </msub>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mi>l</mi>
            <mo stretchy="false">−</mo>
            <msub>
             <mi>x</mi>
             <mn>1</mn>
            </msub>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">T(h)
{dy}
over
{ds}
left(
x_2
right)
= 
%rho g
left(
l - x_1
right)
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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    </draw:frame></text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">ρ</mi>
         <mi>g</mi>
         <mrow>
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          <mrow>
           <mrow>
            <mi>h</mi>
            <mo stretchy="false">+</mo>
            <mi>C</mi>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
         <mfrac>
          <mi mathvariant="italic">dy</mi>
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         </mfrac>
         <mrow>
          <mrow>
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            </msub>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mi>l</mi>
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            <msub>
             <mi>x</mi>
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            </msub>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%rho g 
left(
h + C
right)
{dy}
over
{ds}
left(
x_2
right)
= 
%rho g
left(
l - x_1
right)
</annotation>
       </semantics>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">When the anchor line is taut, the unknowns are:</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>C</mi>
         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>1</mn>
         </msub>
         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>2</mn>
         </msub>
         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>3</mn>
         </msub>
         <mi>,</mi>
         <msub>
          <mi>C</mi>
          <mn>4</mn>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">C,C_1,C_2,C_3,C_4</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AgAXAAAA/ggAAHIBAAABAEMXAQAAAAD//wEAQwCMAAEAAAAAAIsAAQACAAAAHwCKAAEAOwAA
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       AAA=
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">The boundary conditions are:</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>s</mi>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mn>0</mn>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">s left(0 right) = 0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:p text:style-name="Standard">Summing forces in y for the catenary portion:</text:p>
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right)
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left(
0
right)
= 
%rho g
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left(
h + C
right)
{dy}
over
{ds}
left(
x_2
right)
-
%rho g C
{dy}
over
{ds}
left(
0
right)
= 
%rho g l</annotation>
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   <text:p text:style-name="Standard">Solving for the unknowns </text:p>
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              <mrow>
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        </mrow>
        <annotation encoding="StarMath 5.0">C_3
sinh
left(
{
x_1
+ C_5}
over 
{C_4}
right)
-
C_2
-
x_1
=0
</annotation>
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sinh
left(
{
x_2
+ C_5}
over 
{C_4}
right)
-
C_2
-
l
=0
</annotation>
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        <annotation encoding="StarMath 5.0">C_3
cosh
left(
{
x_1
+ C_5}
over 
{C_4}
right)
-
C
=
0</annotation>
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over 
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C
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h
=
0</annotation>
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sinh
left(
{
x_1
+ C_5}
over 
{C_4}
right)
=
0
</annotation>
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{dy}
over
{ds}
left(
x_2
right)
= 
%rho g
left(
l - x_1
right)
</annotation>
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left(
h + C
right)
sinh
left(
{
x_2
+ C_5}
over 
{C_4}
right)
}
over
{
sqrt
{
C^2_3
+
sinh^2
left(
{
x_2
+ C_5}
over 
{C_4}
right)
}
}
- 
%rho g
left(
l - x_1
right)
=
0</annotation>
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sinh
left(
{
0
+ C_5}
over 
{C_4}
right)
-
C_2
-
0
=0
</annotation>
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C_2
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l
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over 
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right)
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C
=
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cosh
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x_2
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C
-
h
=
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over
{ds}
left(
x_2
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{dy}
over
{ds}
left(
0
right)
= 
%rho g
l</annotation>
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                </msub>
               </mfrac>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
            </mrow>
           </msqrt>
          </mfrac>
          <mo stretchy="false">−</mo>
          <mi mathvariant="normal">ρ</mi>
         </mrow>
         <mi>g</mi>
         <mrow>
          <mi>l</mi>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">
{%rho g 
left(
h + C
right)
sinh
left(
{
x_2
+ C_5}
over 
{C_4}
right)
}
over
{
sqrt
{
C^2_3
+
sinh^2
left(
{
x_2
+ C_5}
over 
{C_4}
right)
}
}
+

{%rho g 
left(
0 + C
right)
sinh
left(
{
0
+ C_5}
over 
{C_4}
right)
}
over
{
sqrt
{
C^2_3
+
sinh^2
left(
{
0
+ C_5}
over 
{C_4}
right)
}
}
- 
%rho g l
=
0</annotation>
       </semantics>
      </math>
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