Question Number 81804 by ajfour last updated on 15/Feb/20
Commented by ajfour last updated on 15/Feb/20
$${If}\:{tension}\:{in}\:{string},\:{T}=\frac{{mg}}{\mathrm{5}} \\ $$$${and}\:{rod}\:{rests}\:{in}\:{equilibrium}, \\ $$$${find}\:{length}\:{of}\:{string}. \\ $$
Answered by mr W last updated on 15/Feb/20
Commented by mr W last updated on 16/Feb/20
$$\beta+\theta=\frac{\pi}{\mathrm{2}}−\alpha \\ $$$$\Rightarrow\beta=\frac{\pi}{\mathrm{2}}−\left(\theta+\alpha\right) \\ $$$$\gamma+\theta=\frac{\pi}{\mathrm{2}}+\alpha \\ $$$$\Rightarrow\gamma=\frac{\pi}{\mathrm{2}}−\left(\theta−\alpha\right) \\ $$$$ \\ $$$$\frac{{BD}}{\mathrm{sin}\:\gamma}=\frac{{DC}}{\mathrm{sin}\:\beta}=\frac{{BC}}{\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${BD}=\frac{{L}\:\mathrm{sin}\:\gamma}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta}=\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${DC}=\frac{{L}\:\mathrm{sin}\:\beta}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta}=\frac{{L}\:\mathrm{cos}\:\left(\theta+\alpha\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${l}={BD}+{DC}=\frac{{L}\left[\mathrm{cos}\:\left(\theta−\alpha\right)+\mathrm{cos}\:\left(\theta+\alpha\right)\right]}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\Rightarrow{l}=\frac{{L}\:\mathrm{cos}\:\alpha}{\mathrm{2}\:\mathrm{sin}\:\theta}\:\:\:…\left({iii}\right) \\ $$$$ \\ $$$${BD}\:\mathrm{sin}\:\theta={e}\:\mathrm{cos}\:\alpha \\ $$$$\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha\right)\:\mathrm{sin}\:\theta}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta}={e}\:\mathrm{cos}\:\alpha \\ $$$${e}=\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha\right)}{\mathrm{4}\:\mathrm{cos}\:\theta\:\mathrm{cos}\:\alpha}=\frac{{L}}{\mathrm{4}}\left(\mathrm{1}+\mathrm{tan}\:\theta\:\mathrm{tan}\:\alpha\right) \\ $$$$ \\ $$$${let}\:{T}=\frac{{mg}}{\tau}\:{with}\:\tau=\mathrm{5}\:{given} \\ $$$${F}=\mathrm{2}{T}\:\mathrm{cos}\:\theta \\ $$$${F}\left(\frac{{L}}{\mathrm{2}}+{e}\right)={mg}×\frac{{L}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{3}\:\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:\mathrm{tan}\:\alpha=\tau\:\:\:…\left({i}\right) \\ $$$$ \\ $$$$\frac{{L}\:\mathrm{sin}\:\alpha}{\mathrm{2}}+{BD}×\mathrm{cos}\:\theta={h} \\ $$$$\mathrm{2}\:\mathrm{sin}\:\alpha+\frac{\mathrm{cos}\:\left(\theta−\alpha\right)}{\mathrm{sin}\:\theta}=\frac{\mathrm{4}{h}}{{L}} \\ $$$${with}\:\eta=\frac{{h}}{{L}} \\ $$$$\Rightarrow\mathrm{3}\:\mathrm{sin}\:\alpha+\frac{\mathrm{cos}\:\alpha}{\mathrm{tan}\:\theta}=\mathrm{4}\eta\:\:\:…\left({ii}\right) \\ $$$$ \\ $$$${from}\:\left({i}\right): \\ $$$$\frac{\mathrm{3}}{\:\sqrt{\mathrm{9}+\mathrm{tan}^{\mathrm{2}} \:\alpha}}\:\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:\frac{\mathrm{tan}\:\alpha}{\:\sqrt{\mathrm{9}+\mathrm{tan}^{\mathrm{2}} \:\alpha}}=\frac{\tau}{\:\sqrt{\mathrm{9}+\mathrm{tan}^{\mathrm{2}} \:\alpha}} \\ $$$$\mathrm{cos}\:\left(\theta−\phi\right)=\frac{\tau}{\:\sqrt{\mathrm{9}+\mathrm{tan}^{\mathrm{2}} \:\alpha}}\:{with}\:\mathrm{tan}\:\phi=\frac{\mathrm{tan}\:\alpha}{\mathrm{3}} \\ $$$$\Rightarrow\theta=\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{tan}\:\alpha}{\mathrm{3}}\right)+\mathrm{cos}^{−\mathrm{1}} \left(\frac{\tau}{\:\sqrt{\mathrm{9}+\mathrm{tan}^{\mathrm{2}} \:\alpha}}\right) \\ $$$${put}\:{this}\:{into}\:\left({ii}\right)\:{we}\:{can}\:{get}\:\alpha. \\ $$$$ \\ $$$${with}\:\alpha\:{and}\:\theta,\:{we}\:{get}\:{the}\:{length}\:{of} \\ $$$${the}\:{string}\:{acc}.\:{to}\:\left({iii}\right):\:{l}=\frac{{L}\:\mathrm{cos}\:\alpha}{\mathrm{2}\:\mathrm{sin}\:\theta}. \\ $$$$ \\ $$$${example}:\:\tau=\frac{{mg}}{{T}}=\mathrm{3},\:\eta=\frac{{h}}{{L}}=\frac{\mathrm{4}}{\mathrm{5}}=\mathrm{0}.\mathrm{80} \\ $$$$\Rightarrow\alpha=\mathrm{39}.\mathrm{6988}° \\ $$$$\Rightarrow\theta=\mathrm{30}.\mathrm{9364}° \\ $$$$\Rightarrow{l}=\mathrm{3}.\mathrm{7416}{m} \\ $$$$=================== \\ $$$${if}\:{string}\:{length}\:{l}\:{is}\:{given},\:{find}\:{T}=? \\ $$$${from}\:\left({iii}\right)\:{we}\:{get} \\ $$$$\frac{\mathrm{cos}\:\alpha}{\mathrm{sin}\:\theta}=\frac{\mathrm{2}{l}}{{L}}=\lambda \\ $$$$\Rightarrow\mathrm{sin}\:\theta=\frac{\mathrm{cos}\:\alpha}{\lambda} \\ $$$${put}\:{this}\:{into}\:\left({ii}\right): \\ $$$$\mathrm{3}\:\mathrm{sin}\:\alpha+\sqrt{\lambda^{\mathrm{2}} −\mathrm{cos}^{\mathrm{2}} \:\alpha}=\mathrm{4}\eta \\ $$$$\sqrt{\lambda^{\mathrm{2}} −\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \:\alpha}=\mathrm{4}\eta−\mathrm{3}\:\mathrm{sin}\:\alpha \\ $$$$\mathrm{8}\:\mathrm{sin}^{\mathrm{2}} \:\alpha−\mathrm{24}\eta\:\mathrm{sin}\:\alpha+\mathrm{16}\eta^{\mathrm{2}} +\mathrm{1}−\lambda^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\mathrm{sin}\:\alpha=\frac{\mathrm{6}\eta−\sqrt{\mathrm{2}\left(\mathrm{2}\eta^{\mathrm{2}} +\lambda^{\mathrm{2}} −\mathrm{1}\right)}}{\mathrm{4}} \\ $$$${with}\:\alpha\:{and}\:\theta\:{we}\:{can}\:{get}\:{from}\:\left({i}\right): \\ $$$$\tau=\mathrm{3}\:\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:\mathrm{tan}\:\alpha \\ $$$$\Rightarrow\tau=\frac{\mathrm{1}}{\lambda}\left(\mathrm{3}\sqrt{\lambda^{\mathrm{2}} −\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \:\alpha}+\mathrm{sin}\:\alpha\right) \\ $$$$ \\ $$$${example}:\:{L}=\mathrm{4}{m},\:{h}=\mathrm{3}{m},\:{l}=\mathrm{3}{m} \\ $$$$\eta=\frac{{h}}{{L}}=\frac{\mathrm{3}}{\mathrm{4}}=\mathrm{0}.\mathrm{75},\:\lambda=\frac{\mathrm{2}{l}}{{L}}=\frac{\mathrm{6}}{\mathrm{4}}=\mathrm{1}.\mathrm{5} \\ $$$$\mathrm{sin}\:\alpha=\frac{\mathrm{6}\eta−\sqrt{\mathrm{2}\left(\mathrm{2}\eta^{\mathrm{2}} +\lambda^{\mathrm{2}} −\mathrm{1}\right)}}{\mathrm{4}}=\mathrm{0}.\mathrm{580137} \\ $$$$\Rightarrow\tau=\mathrm{2}.\mathrm{905932} \\ $$$$\Rightarrow{T}=\frac{{mg}}{\tau}=\mathrm{0}.\mathrm{344}{mg} \\ $$
Commented by ajfour last updated on 15/Feb/20
$${great}\:{way}\:{Sir},\:{its}\:{really}\:{not}\:{so} \\ $$$${complex},\:{thank}\:{you},\:{i}\:{think} \\ $$$${my}\:{solution}\:{answers}\:{well}\:{too}. \\ $$
Commented by mr W last updated on 15/Feb/20
$${that}'{s}\:{sure}\:{sir}!\:{your}\:{solution}\:{solves} \\ $$$${the}\:{problem}\:{also}\:{very}\:{well}.\:{many} \\ $$$${ways}\:{lead}\:{to}\:{the}\:{same}\:{goal}! \\ $$
Commented by mr W last updated on 16/Feb/20
$${for}\:{the}\:{case}:\:{l}\:{is}\:{given},\:{find}\:{T}=? \\ $$$${an}\:{exact}\:{solution}\:{is}\:{possible},\:{see}\:{above}. \\ $$
Answered by ajfour last updated on 15/Feb/20
Commented by ajfour last updated on 15/Feb/20
$${I}\:{am}\:{enlisting}\:{just}\:{the}\:{eqs}. \\ $$$${that}\:{i}\:{have}: \\ $$$$\:\:\:\:{h}−\frac{{L}\:}{\mathrm{2}}\mathrm{sin}\:\theta={p}\mathrm{cos}\:\theta\:\:\:\:….\left({i}\right) \\ $$$$\:\:\:\:{h}−{L}\mathrm{sin}\:\theta={q}\mathrm{cos}\:\theta\:\:\:\:\:\:…\left({ii}\right) \\ $$$$\:\:\:\:{p}\mathrm{cos}\:\alpha={q}\mathrm{cos}\:\beta \\ $$$$\:\:\:\:{p}\mathrm{sin}\:\alpha+{q}\mathrm{sin}\:\beta=\frac{{L}}{\mathrm{2}} \\ $$$$\:\:\:\:\mathrm{cos}\:\alpha+\mathrm{2cos}\:\beta=\mathrm{5cos}\:\theta \\ $$$$\:\:\:\:\:\:\:\:\:\:\alpha−\beta\:=\:\mathrm{2}\theta \\ $$$$\left({one}\:{of}\:{them}\:{is}\:{is}\:{dependent}\right) \\ $$$$\mathrm{cos}\:\alpha+\frac{\mathrm{2}{p}\mathrm{cos}\:\alpha}{{q}}=\mathrm{5cos}\:\theta \\ $$$$\Rightarrow\:\:\:\mathrm{cos}\:\alpha=\frac{\mathrm{5cos}\:\theta}{\left(\mathrm{1}+\mathrm{2}{p}/{q}\right)} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{cos}\:\beta=\frac{\mathrm{5}\left({p}/{q}\right)\mathrm{cos}\:\theta}{\left(\mathrm{1}+\mathrm{2}{p}/{q}\right)} \\ $$$$ \\ $$$$\:\:{p}\sqrt{\mathrm{1}−\left\{\frac{\mathrm{5cos}\:\theta}{\left(\mathrm{1}+\mathrm{2}{p}/{q}\right)}\right\}^{\mathrm{2}} }+ \\ $$$$\:\:\:\:{q}\sqrt{\mathrm{1}−\left\{\frac{\mathrm{5}\left({p}/{q}\right)\mathrm{cos}\:\theta}{\left(\mathrm{1}+\mathrm{2}{p}/{q}\right)}\right\}^{\mathrm{2}} }\:=\frac{{L}}{\mathrm{2}} \\ $$$${while}\:{we}\:{have}\:{p}\left(\theta\right)\:{and}\:{q}\left(\theta\right), \\ $$$${from}\:{first}\:{two}\:{eqs}. \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$