Mathematical curves and Hooke’s chain theory
W 0 : weight of each unit length of the string; S: the length of the section, so the total weight can be expressed as W 0 S; T: the tension exerted on the string; T 0 : the load exerted on the section from the negative x part of the curve.
Figure 2 Force analysis of said string segment
It is known that the string is in equilibrium as the system is stationary, which means that ∑𝐹 𝑥 =0 and ∑𝐹 𝑦 =0
Balancing the horizontal forces:
𝑇 0 =𝑇cos𝜃 (1) And assuming that the length of the section is infinitesimal as its y length is 𝑑𝑦 and x length is 𝑑𝑥 :
𝑑𝑦 𝑑𝑥
tan𝜃 =
(2)
And balancing the vertical forces:
𝑊 0 𝑆=𝑇sin𝜃 (3)
Divide (3) by (1):
𝑊 0 𝑆 𝑇 0
= tan𝜃(4),𝑤ℎ𝑖𝑐ℎ,𝑖𝑛 (2),𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑑𝑦 𝑑𝑥
𝑊 0 𝑇 0
is a constant, let k =
, so
𝑑𝑦 𝑑𝑥
= 𝑘𝑠 (5)
the arclength formula is that 𝑠 =∫√1+ 𝑑𝑦 𝑑𝑥 2 𝑑𝑥 ,
Differentiate both sides of the arclength formula,
2
𝑑𝑠 𝑑𝑥
𝑑𝑦 𝑑𝑥
=√1+
(6)
then differentiate (5) with respect to x, then substitute 𝑑𝑠 𝑑𝑥
with the relation in (6)
2
𝑑 2 𝑦 𝑑𝑥 2
𝑑𝑠 𝑑𝑥
𝑑𝑦 𝑑𝑥
= 𝑘
= 𝑘√1+
(7)
Now, for the ease of expression, let 𝑝 = 𝑑𝑦 𝑑𝑥
, substitute in (7),
𝑑𝑝 𝑑𝑥
= 𝑘√1+𝑝 2 (8)
Separate variables in (8):
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