Question Number 193719 by mr W last updated on 18/Jun/23
Commented by mr W last updated on 18/Jun/23
$${the}\:{thin}\:{rope}\:{consists}\:{of}\:{two}\:{uniform} \\ $$$${parts}: \\ $$$${part}\:\mathrm{1}:\:{length}\:{l}_{\mathrm{1}} ,\:{mass}\:{m}_{\mathrm{1}} \\ $$$${part}\:\mathrm{2}:\:{length}\:{l}_{\mathrm{2}} ,\:{mass}\:{m}_{\mathrm{2}} \\ $$$${assume}\:{l}_{\mathrm{2}} >{l}_{\mathrm{1}} ,\:{m}_{\mathrm{2}} >{m}_{\mathrm{1}} . \\ $$$${find}\:{the}\:{position}\:{of}\:{the}\:{lowest}\:{point}\: \\ $$$${of}\:{the}\:{rope}. \\ $$
Answered by mr W last updated on 18/Jun/23
Commented by mr W last updated on 25/Jun/23
![ρ_1 =(m_1 /l_1 ) ρ_2 =(m_2 /l_2 ) a_1 =(T_0 /(ρ_1 g))=((l_1 T_0 )/(m_1 g)) a_2 =(T_0 /(ρ_2 g))=((l_2 T_0 )/(m_2 g)) y_B =a_1 cosh (x_B /a_1 ) y_C =y_B +d=a_1 cosh ((x_B +b_1 )/a_1 ) d=a_1 (cosh ((x_B +b_1 )/a_1 )−cosh (x_B /a_1 )) l_1 =a_1 (sinh ((x_B +b_1 )/a_1 )−sinh (x_B /a_1 )) tan ϕ=sinh (x_B /a_1 ) ⇒x_B =a_1 sinh^(−1) tan ϕ=ka_1 ⇒cosh (k+(b_1 /a_1 ))−cosh k=(d/a_1 ) ⇒sinh (k+(b_1 /a_1 ))−sinh k=(l_1 /a_1 ) similarly ⇒cosh (k+(b_2 /a_2 ))−cosh k=(d/a_2 ) ⇒sinh (k+(b_2 /a_2 ))+sinh k=(l_2 /a_2 ) b_1 +b_2 =b as unknowns we have d, k, T_0 , b_1 , b_2 (b_1 /a_1 )=cosh^(−1) (cosh k+(d/a_1 ))−k (b_1 /a_1 )=sinh^(−1) (sinh k+(l_1 /a_1 ))−k ⇒sinh^(−1) (sinh k+(l_1 /a_1 ))=cosh^(−1) (cosh k+(d/a_1 )) (d/a_1 )=(√(1+(sinh k+(l_1 /a_1 ))^2 ))−cosh k ⇒sinh^(−1) (−sinh k+(l_2 /a_2 ))=cosh^(−1) (cosh k+(d/a_2 )) (d/a_2 )=(√(1+(−sinh k+(l_2 /a_2 ))^2 ))−cosh k ⇒a_1 [(√(1+(sinh k+(l_1 /a_1 ))^2 ))−cosh k]=a_2 [(√(1+(−sinh k+(l_2 /a_2 ))^2 ))−cosh k] ...(i) ⇒a_1 [cosh^(−1) (√(1+(sinh k+(l_1 /a_1 ))^2 ))−k]+a_2 [cosh^(−1) (√(1+(−sinh k+(l_2 /a_2 ))^2 ))−k]=b ...(ii)](https://www.tinkutara.com/question/Q193989.png)
$$\rho_{\mathrm{1}} =\frac{{m}_{\mathrm{1}} }{{l}_{\mathrm{1}} } \\ $$$$\rho_{\mathrm{2}} =\frac{{m}_{\mathrm{2}} }{{l}_{\mathrm{2}} } \\ $$$${a}_{\mathrm{1}} =\frac{{T}_{\mathrm{0}} }{\rho_{\mathrm{1}} {g}}=\frac{{l}_{\mathrm{1}} {T}_{\mathrm{0}} }{{m}_{\mathrm{1}} {g}} \\ $$$${a}_{\mathrm{2}} =\frac{{T}_{\mathrm{0}} }{\rho_{\mathrm{2}} {g}}=\frac{{l}_{\mathrm{2}} {T}_{\mathrm{0}} }{{m}_{\mathrm{2}} {g}} \\ $$$${y}_{{B}} ={a}_{\mathrm{1}} \:\mathrm{cosh}\:\frac{{x}_{{B}} }{{a}_{\mathrm{1}} } \\ $$$${y}_{{C}} ={y}_{{B}} +{d}={a}_{\mathrm{1}} \:\mathrm{cosh}\:\frac{{x}_{{B}} +{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} } \\ $$$${d}={a}_{\mathrm{1}} \:\left(\mathrm{cosh}\:\frac{{x}_{{B}} +{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }−\mathrm{cosh}\:\frac{{x}_{{B}} }{{a}_{\mathrm{1}} }\right) \\ $$$${l}_{\mathrm{1}} ={a}_{\mathrm{1}} \:\left(\mathrm{sinh}\:\frac{{x}_{{B}} +{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }−\mathrm{sinh}\:\frac{{x}_{{B}} }{{a}_{\mathrm{1}} }\right) \\ $$$$\mathrm{tan}\:\varphi=\mathrm{sinh}\:\frac{{x}_{{B}} }{{a}_{\mathrm{1}} } \\ $$$$\Rightarrow{x}_{{B}} ={a}_{\mathrm{1}} \:\mathrm{sinh}^{−\mathrm{1}} \:\mathrm{tan}\:\varphi={ka}_{\mathrm{1}} \\ $$$$\Rightarrow\mathrm{cosh}\:\left({k}+\frac{{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)−\mathrm{cosh}\:{k}=\frac{{d}}{{a}_{\mathrm{1}} } \\ $$$$\Rightarrow\mathrm{sinh}\:\left({k}+\frac{{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)−\mathrm{sinh}\:{k}=\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} } \\ $$$${similarly} \\ $$$$\Rightarrow\mathrm{cosh}\:\left({k}+\frac{{b}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)−\mathrm{cosh}\:{k}=\frac{{d}}{{a}_{\mathrm{2}} } \\ $$$$\Rightarrow\mathrm{sinh}\:\left({k}+\frac{{b}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)+\mathrm{sinh}\:{k}=\frac{{l}_{\mathrm{2}} }{{a}_{\mathrm{2}} } \\ $$$${b}_{\mathrm{1}} +{b}_{\mathrm{2}} ={b} \\ $$$${as}\:{unknowns}\:{we}\:{have} \\ $$$${d},\:{k},\:{T}_{\mathrm{0}} ,\:{b}_{\mathrm{1}} ,\:{b}_{\mathrm{2}} \\ $$$$\frac{{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }=\mathrm{cosh}^{−\mathrm{1}} \:\left(\mathrm{cosh}\:{k}+\frac{{d}}{{a}_{\mathrm{1}} }\right)−{k} \\ $$$$\frac{{b}_{\mathrm{1}} }{{a}_{\mathrm{1}} }=\mathrm{sinh}^{−\mathrm{1}} \:\left(\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)−{k} \\ $$$$\Rightarrow\mathrm{sinh}^{−\mathrm{1}} \:\left(\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)=\mathrm{cosh}^{−\mathrm{1}} \:\left(\mathrm{cosh}\:{k}+\frac{{d}}{{a}_{\mathrm{1}} }\right) \\ $$$$\frac{{d}}{{a}_{\mathrm{1}} }=\sqrt{\mathrm{1}+\left(\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)^{\mathrm{2}} }−\mathrm{cosh}\:{k} \\ $$$$\Rightarrow\mathrm{sinh}^{−\mathrm{1}} \:\left(−\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)=\mathrm{cosh}^{−\mathrm{1}} \:\left(\mathrm{cosh}\:{k}+\frac{{d}}{{a}_{\mathrm{2}} }\right) \\ $$$$\frac{{d}}{{a}_{\mathrm{2}} }=\sqrt{\mathrm{1}+\left(−\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)^{\mathrm{2}} }−\mathrm{cosh}\:{k} \\ $$$$\Rightarrow{a}_{\mathrm{1}} \left[\sqrt{\mathrm{1}+\left(\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)^{\mathrm{2}} }−\mathrm{cosh}\:{k}\right]={a}_{\mathrm{2}} \left[\sqrt{\mathrm{1}+\left(−\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)^{\mathrm{2}} }−\mathrm{cosh}\:{k}\right]\:\:\:…\left({i}\right) \\ $$$$\Rightarrow{a}_{\mathrm{1}} \left[\mathrm{cosh}^{−\mathrm{1}} \:\sqrt{\mathrm{1}+\left(\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{1}} }{{a}_{\mathrm{1}} }\right)^{\mathrm{2}} }−{k}\right]+{a}_{\mathrm{2}} \left[\mathrm{cosh}^{−\mathrm{1}} \:\sqrt{\mathrm{1}+\left(−\mathrm{sinh}\:{k}+\frac{{l}_{\mathrm{2}} }{{a}_{\mathrm{2}} }\right)^{\mathrm{2}} }−{k}\right]={b}\:\:\:…\left({ii}\right) \\ $$