find the intrinsic equation of the catenary y=C cosh x/C.
Hello.
The catenary is a plane curve, whose shape corresponds to a hanging homogeneous flexible chain supported at its ends and sagging under the force of gravity.
Let,
( x,y ) be the arbitrary point on the catenary.
s be the length of arc from the lowest point of catenary to the point ( x, y )
wo be the linear density of the catenary.
So, weight of gravity = wos
To be the tension at lowest point
T1 be the tension at ( x, y )
Hence,
Resolving the tension forces in X and Y directions, we get, the tension in X directions,
To = T cosθ.......... ( 1 )
wos = T sin θ........ ( 2 )
Dividing ( 2 ) by ( 1 ) we get,
tan θ = wos / To
So, s = To . tan θ / wo
Let, To / wo = c
Hence, s = c. tanθ........ ( 3 )
The above equation ( 3 ) is known as the intrinsic equation of the Catenary.
Best Wishes.
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