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Question Number 193719 by mr W last updated on 18/Jun/23

Commented by mr W last updated on 18/Jun/23

$t h e t h i n r o p e c o n s i s t s o f t w o u n i f o r m$ $p a r t s :$ $p a r t 1 : l e n g t h l_{1}, m a s s m_{1}$ $p a r t 2 : l e n g t h l_{2}, m a s s m_{2}$ $a s s u m e l_{2} > l_{1}, m_{2} > m_{1} .$ $f i n d t h e p o s i t i o n o f t h e l o w e s t p o i n t$ $o f t h e r o p e .$
	Answered by mr W last updated on 18/Jun/23

	Commented by mr W last updated on 25/Jun/23

	$ρ_{1} = \frac{m_{1}}{l_{1}}$ $ρ_{2} = \frac{m_{2}}{l_{2}}$ $a_{1} = \frac{T_{0}}{ρ_{1} g} = \frac{l_{1} T_{0}}{m_{1} g}$ $a_{2} = \frac{T_{0}}{ρ_{2} g} = \frac{l_{2} T_{0}}{m_{2} g}$ $y_{B} = a_{1} \cosh \frac{x_{B}}{a_{1}}$ $y_{C} = y_{B} + d = a_{1} \cosh \frac{x_{B} + b_{1}}{a_{1}}$ $d = a_{1} (\cosh \frac{x_{B} + b_{1}}{a_{1}} - \cosh \frac{x_{B}}{a_{1}})$ $l_{1} = a_{1} (\sinh \frac{x_{B} + b_{1}}{a_{1}} - \sinh \frac{x_{B}}{a_{1}})$ $\tan φ = \sinh \frac{x_{B}}{a_{1}}$ $\Rightarrow x_{B} = a_{1} \sinh^{- 1} \tan φ = {k a}_{1}$ $\Rightarrow \cosh (k + \frac{b_{1}}{a_{1}}) - \cosh k = \frac{d}{a_{1}}$ $\Rightarrow \sinh (k + \frac{b_{1}}{a_{1}}) - \sinh k = \frac{l_{1}}{a_{1}}$ $s i m i l a r l y$ $\Rightarrow \cosh (k + \frac{b_{2}}{a_{2}}) - \cosh k = \frac{d}{a_{2}}$ $\Rightarrow \sinh (k + \frac{b_{2}}{a_{2}}) + \sinh k = \frac{l_{2}}{a_{2}}$ $b_{1} + b_{2} = b$ $a s u n k n o w n s w e h a v e$ $d, k, T_{0}, b_{1}, b_{2}$ $\frac{b_{1}}{a_{1}} = \cosh^{- 1} (\cosh k + \frac{d}{a_{1}}) - k$ $\frac{b_{1}}{a_{1}} = \sinh^{- 1} (\sinh k + \frac{l_{1}}{a_{1}}) - k$ $\Rightarrow \sinh^{- 1} (\sinh k + \frac{l_{1}}{a_{1}}) = \cosh^{- 1} (\cosh k + \frac{d}{a_{1}})$ $\frac{d}{a_{1}} = \sqrt{1 + {(\sinh k + \frac{l_{1}}{a_{1}})}^{2}} - \cosh k$ $\Rightarrow \sinh^{- 1} (- \sinh k + \frac{l_{2}}{a_{2}}) = \cosh^{- 1} (\cosh k + \frac{d}{a_{2}})$ $\frac{d}{a_{2}} = \sqrt{1 + {(- \sinh k + \frac{l_{2}}{a_{2}})}^{2}} - \cosh k$ $\Rightarrow a_{1} [\sqrt{1 + {(\sinh k + \frac{l_{1}}{a_{1}})}^{2}} - \cosh k] = a_{2} [\sqrt{1 + {(- \sinh k + \frac{l_{2}}{a_{2}})}^{2}} - \cosh k] . . . (i)$ $\Rightarrow a_{1} [\cosh^{- 1} \sqrt{1 + {(\sinh k + \frac{l_{1}}{a_{1}})}^{2}} - k] + a_{2} [\cosh^{- 1} \sqrt{1 + {(- \sinh k + \frac{l_{2}}{a_{2}})}^{2}} - k] = b . . . (i i)$
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