Question Number 81968 by mr W last updated on 17/Feb/20
Commented by mr W last updated on 17/Feb/20
$$\left[{derived}\:{from}\:{Q}\mathrm{81804}\right] \\ $$$$ \\ $$$${given}: \\ $$$${string}\:{length}\:={s}\:\:\:\left(>\:\mathrm{0}.\mathrm{5}{L}\right) \\ $$$${friction}\:{coefficient}\:{with}\:{ground}\:=\mu \\ $$$$ \\ $$$${to}\:{find}: \\ $$$${the}\:{minimal}\:{and}\:{maximal}\:{possible} \\ $$$${inclination}\:{of}\:{the}\:{rod}\:{and}\:{the}\:\: \\ $$$${corresponding}\:{tension}\:{in}\:{the}\:{string}. \\ $$
Answered by mr W last updated on 27/Feb/20
Commented by mr W last updated on 27/Feb/20
$${Case}\:{I}:\:{maximal}\:{inclination} \\ $$$${with}\:{friction}\:{force}\:{acting}\:{rightwards} \\ $$$$\varphi=\mathrm{tan}^{−\mathrm{1}} \mu \\ $$$${both}\:{reactions}\:{from}\:{ground}\:{and}\:{from} \\ $$$${pulley}\:{meet}\:{mg}\:{at}\:{the}\:{same}\:{point}\:{O}. \\ $$$$\beta=\frac{\pi}{\mathrm{2}}−\left(\theta+\alpha−\phi\right) \\ $$$$\gamma=\frac{\pi}{\mathrm{2}}−\left(\theta−\alpha+\phi\right) \\ $$$$\frac{{BC}}{\mathrm{sin}\:\gamma}=\frac{{CD}}{\mathrm{sin}\:\beta}=\frac{{BD}}{\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\Rightarrow{BC}=\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha+\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\Rightarrow{CD}=\frac{{L}\:\mathrm{cos}\:\left(\theta+\alpha−\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${BC}+{CD}={s} \\ $$$$\Rightarrow\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha+\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta}+\frac{{L}\:\mathrm{cos}\:\left(\theta+\alpha−\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta}={s} \\ $$$$\Rightarrow{L}\:\mathrm{cos}\:\theta\:\mathrm{cos}\:\left(\alpha−\phi\right)={s}\:\mathrm{sin}\:\mathrm{2}\theta \\ $$$$\Rightarrow\mathrm{sin}\:\theta=\frac{{L}\:\mathrm{cos}\:\left(\alpha−\phi\right)}{\mathrm{2}{s}} \\ $$$${let}\:\Theta=\alpha−\phi \\ $$$$\Rightarrow\mathrm{sin}\:\theta=\frac{{L}\:\mathrm{cos}\:\Theta}{\mathrm{2}{s}}\:\:\:…\left({i}\right) \\ $$$$ \\ $$$${h}=\frac{{L}\:\mathrm{sin}\:\alpha}{\mathrm{2}}+{BC}×\mathrm{cos}\:\left(\theta−\phi\right) \\ $$$${h}=\frac{{L}\:\mathrm{sin}\:\alpha}{\mathrm{2}}+\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha+\phi\right)\:\mathrm{cos}\:\left(\theta−\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\frac{\mathrm{2}{h}}{{L}}=\mathrm{sin}\:\alpha+\frac{\mathrm{cos}\:\left(\theta−\alpha+\phi\right)\:\mathrm{cos}\:\left(\theta−\phi\right)}{\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\frac{\mathrm{2}{h}}{{L}}=\mathrm{sin}\:\alpha+\frac{\left[\mathrm{cos}\:\theta\:\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{sin}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\left(\mathrm{cos}\:\theta\:\mathrm{cos}\:\phi+\mathrm{sin}\:\theta\:\mathrm{sin}\:\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta} \\ $$$$\Rightarrow\frac{\mathrm{4}{h}}{{L}}=\mathrm{2}\:\mathrm{sin}\:\alpha+\left[\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{tan}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\left(\frac{\mathrm{cos}\:\phi}{\mathrm{tan}\:\theta}+\mathrm{sin}\:\phi\right) \\ $$$$\Rightarrow\frac{\mathrm{4}{h}}{{L}}=\mathrm{2}\:\mathrm{sin}\:\alpha+\left(\mathrm{cos}\:\Theta+\mathrm{tan}\:\theta\:\mathrm{sin}\:\Theta\right)\left[\frac{\mathrm{cos}\:\left(\alpha−\Theta\right)}{\mathrm{tan}\:\theta}+\mathrm{sin}\:\left(\alpha−\Theta\right)\right]\:\:\:…\left({ii}\right) \\ $$$$ \\ $$$${GC}={BC}×\mathrm{sin}\:\left(\theta−\phi\right) \\ $$$${GC}=\frac{{L}\:\mathrm{cos}\:\left(\theta−\alpha+\phi\right)\:\mathrm{sin}\:\left(\theta−\phi\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${GC}=\frac{{L}\:\left[\mathrm{cos}\:\theta\:\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{sin}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\:\left(\mathrm{sin}\:\theta\:\mathrm{cos}\:\phi−\mathrm{cos}\:\theta\:\mathrm{sin}\:\phi\right)}{\mathrm{4}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta} \\ $$$${GC}=\frac{{L}\left[\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{tan}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\:\left(\mathrm{cos}\:\phi−\frac{\mathrm{sin}\:\phi}{\mathrm{tan}\:\theta}\right)}{\mathrm{4}} \\ $$$${OG}=\frac{{L}\:\mathrm{cos}\:\alpha}{\mathrm{2}\:\mathrm{tan}\:\varphi}−{h} \\ $$$${OG}×\mathrm{tan}\:\phi={GC} \\ $$$$\left(\frac{{L}\:\mathrm{cos}\:\alpha}{\mathrm{2}\:\mathrm{tan}\:\varphi}−{h}\right)\mathrm{tan}\:\phi=\frac{{L}\left[\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{tan}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\:\left(\mathrm{cos}\:\phi−\frac{\mathrm{sin}\:\phi}{\mathrm{tan}\:\theta}\right)}{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{2}\left(\frac{\mathrm{cos}\:\alpha}{\mu}−\frac{\mathrm{2}{h}}{{L}}\right)\mathrm{tan}\:\phi=\left[\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{tan}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\:\left(\mathrm{cos}\:\phi−\frac{\mathrm{sin}\:\phi}{\mathrm{tan}\:\theta}\right) \\ $$$$\Rightarrow\mathrm{2}\left(\frac{\mathrm{cos}\:\alpha}{\mu}−\frac{\mathrm{2}{h}}{{L}}\right)\mathrm{tan}\:\left(\alpha−\Theta\right)=\left(\mathrm{cos}\:\Theta+\mathrm{tan}\:\theta\:\mathrm{sin}\:\Theta\right)\:\left[\mathrm{cos}\:\left(\alpha−\Theta\right)−\frac{\mathrm{sin}\:\left(\alpha−\Theta\right)}{\mathrm{tan}\:\theta}\right]\:\:\:…\left({iii}\right) \\ $$$${by}\:{putting}\:\left({i}\right)\:{into}\:\left({ii}\right)\:{and}\:\left({iii}\right)\:{we}\:{get} \\ $$$${two}\:{equations}\:{for}\:\alpha\:{and}\:\Theta. \\ $$$$ \\ $$$${example}: \\ $$$${L}=\mathrm{5}{m},\:{h}=\mathrm{3}{m},\:{s}=\mathrm{4}{m},\:\mu=\mathrm{0}.\mathrm{6} \\ $$$$\Rightarrow\alpha=\mathrm{22}.\mathrm{882}° \\ $$$$\Rightarrow\Theta=\alpha−\phi=\mathrm{7}.\mathrm{0125}° \\ $$
Commented by mr W last updated on 28/Feb/20
$$\frac{{BC}}{\mathrm{sin}\:\left(\mathrm{2}\theta+\beta\right)}=\frac{{CD}}{\mathrm{sin}\:\beta}=\frac{{L}}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${BC}=\frac{{L}\:\mathrm{sin}\:\left(\mathrm{2}\theta+\beta\right)}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${CD}=\frac{{L}\:\mathrm{sin}\:\beta}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$${s}=\frac{{L}\:\left[\mathrm{sin}\:\mathrm{2}\theta\:\mathrm{cos}\:\beta+\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta\right)\mathrm{sin}\:\beta\right]}{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\Rightarrow\frac{\mathrm{2}{s}}{{L}}=\mathrm{cos}\:\beta+\frac{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta\right)\mathrm{sin}\:\beta}{\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\Rightarrow\frac{\mathrm{2}{s}}{{L}}×\frac{\mathrm{sin}\:\mathrm{2}\theta}{\:\sqrt{\mathrm{2}\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta\right)}}=\mathrm{sin}\:\lambda\:\mathrm{cos}\:\beta+\mathrm{cos}\:\lambda\:\mathrm{sin}\:\beta \\ $$$$\Rightarrow\frac{\mathrm{2}{s}}{{L}}×\frac{\mathrm{sin}\:\mathrm{2}\theta}{\:\sqrt{\mathrm{2}\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta\right)}}=\mathrm{sin}\:\left(\lambda+\beta\right) \\ $$$$\Rightarrow\beta=\mathrm{sin}^{−\mathrm{1}} \left[\frac{\mathrm{2}{s}}{{L}}×\frac{\mathrm{sin}\:\mathrm{2}\theta}{\:\sqrt{\mathrm{2}\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta\right)}}\right]−\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{1}+\mathrm{cos}\:\mathrm{2}\theta} \\ $$$$ \\ $$$${h}=\frac{{L}\:\mathrm{sin}\:\alpha}{\mathrm{2}}+{BC}×\mathrm{sin}\:\left(\beta+\alpha\right) \\ $$$$\frac{\mathrm{2}{h}}{{L}}=\mathrm{sin}\:\alpha+\frac{\mathrm{sin}\:\left(\mathrm{2}\theta+\beta\right)\:\mathrm{sin}\:\left(\beta+\alpha\right)}{\mathrm{sin}\:\mathrm{2}\theta} \\ $$$$\frac{\mathrm{2}{h}}{{L}}=\mathrm{sin}\:\alpha+\left(\mathrm{cos}\:\beta+\frac{\mathrm{sin}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)\left(\mathrm{sin}\:\beta\:\mathrm{cos}\:\alpha+\mathrm{cos}\:\beta\:\mathrm{sin}\:\alpha\right) \\ $$$$\frac{\mathrm{2}{h}}{{L}}=\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:\beta+\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)\mathrm{sin}\:\alpha+\left(\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta+\frac{\mathrm{sin}^{\mathrm{2}} \:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)\:\mathrm{cos}\:\alpha \\ $$$$\frac{\mathrm{2}{h}}{{L}}×\frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:\beta+\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} +\left(\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta+\frac{\mathrm{sin}^{\mathrm{2}} \:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} }}=\mathrm{cos}\:\delta\:\mathrm{sin}\:\alpha+\mathrm{sin}\:\delta\:\mathrm{cos}\:\alpha \\ $$$$\frac{\mathrm{2}{h}}{{L}}×\frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:\beta+\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} +\left(\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta+\frac{\mathrm{sin}^{\mathrm{2}} \:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} }}=\:\mathrm{sin}\:\left(\alpha+\delta\right) \\ $$$$\Rightarrow\alpha=\mathrm{sin}^{−\mathrm{1}} \left[\frac{\mathrm{2}{h}}{{L}}×\frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:\beta+\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} +\left(\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta+\frac{\mathrm{sin}^{\mathrm{2}} \:\beta}{\mathrm{tan}\:\mathrm{2}\theta}\right)^{\mathrm{2}} }}\right]−\mathrm{tan}^{−\mathrm{1}} \left[\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta+\frac{\mathrm{sin}^{\mathrm{2}} \:\beta}{\mathrm{tan}\:\mathrm{2}\theta}}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:\beta+\frac{\mathrm{sin}\:\beta\:\mathrm{cos}\:\beta}{\mathrm{tan}\:\mathrm{2}\theta}}\right] \\ $$$$\Rightarrow\mathrm{2}\left(\frac{\mathrm{cos}\:\alpha}{\mu}−\frac{\mathrm{2}{h}}{{L}}\right)\mathrm{tan}\:\phi=\left[\mathrm{cos}\:\left(\alpha−\phi\right)+\mathrm{tan}\:\theta\:\mathrm{sin}\:\left(\alpha−\phi\right)\right]\:\left(\mathrm{cos}\:\phi−\frac{\mathrm{sin}\:\phi}{\mathrm{tan}\:\theta}\right) \\ $$$$\phi=\theta+\alpha+\beta−\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{2}\left(\frac{\mathrm{cos}\:\alpha}{\mu}−\frac{\mathrm{2}{h}}{{L}}\right)\frac{\mathrm{1}}{\mathrm{tan}\:\left(\theta+\alpha+\beta\right)}+\left[\mathrm{sin}\:\left(\theta+\beta\right)+\mathrm{tan}\:\theta\:\mathrm{cos}\:\left(\theta+\beta\right)\right]\:\left[\mathrm{sin}\:\left(\theta+\alpha+\beta\right)+\frac{\mathrm{cos}\:\left(\theta+\alpha+\beta\right)}{\mathrm{tan}\:\theta}\right]=\mathrm{0} \\ $$
Commented by ajfour last updated on 29/Feb/20
$$\mathrm{Even}\:\mathrm{with}\:\mathrm{your}\:\mathrm{solution}\:\mathrm{its}\:\mathrm{no} \\ $$$$\mathrm{simpler}\:\mathrm{sir}… \\ $$
Commented by mr W last updated on 29/Feb/20
$${no}\:{sir}! \\ $$$${it}\:{can}\:{not}\:{become}\:{more}\:{simple},\:{i}\:{think}. \\ $$$${but}\:{we}\:{get}\:{a}\:{final}\:{equation}\:{for}\:{a}\:{single} \\ $$$${parameter}\:\theta. \\ $$