Question Number 159085 by ajfour last updated on 12/Nov/21
Answered by mr W last updated on 15/Nov/21
Commented by mr W last updated on 15/Nov/21
$$\mathrm{2}{R}\mathrm{sin}\:\theta={R}+{R}\mathrm{sin}\:\varphi \\ $$$$\mathrm{sin}\:\varphi=\mathrm{2}\:\mathrm{sin}\:\theta−\mathrm{1} \\ $$$$\frac{{d}\varphi}{{d}\theta}=\frac{\mathrm{2}\:\mathrm{cos}\:\theta}{\mathrm{cos}\:\varphi}=\frac{\mathrm{cos}\:\theta}{\:\sqrt{\mathrm{sin}\:\theta\left(\mathrm{1}−\mathrm{sin}\:\theta\right)}} \\ $$$${x}_{{A}} ={x}_{{E}} =\mathrm{4}{R}\:\mathrm{sin}\:\theta \\ $$$${x}_{{B}} ={x}_{{A}} +\mathrm{2}{R}\:\mathrm{cos}\:\theta+{R}\:\mathrm{cos}\:\varphi \\ $$$${x}_{{B}} =\mathrm{4}{R}\:\mathrm{sin}\:\theta+\mathrm{2}{R}\:\mathrm{cos}\:\theta+{R}\:\mathrm{cos}\:\varphi \\ $$$$\omega=\frac{{d}\theta}{{dt}} \\ $$$${v}_{{B}} =\frac{{dx}_{{B}} }{{dt}}=\mathrm{4}{R}\mathrm{cos}\:\theta\:\omega−\mathrm{2}{R}\:\mathrm{sin}\:\theta\:\omega−{R}\:\mathrm{sin}\:\varphi\:\omega\:\frac{{d}\varphi}{{d}\theta} \\ $$$${v}_{{B}} =\mathrm{2}{R}\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)\omega \\ $$$$\frac{{dv}_{{B}} }{{d}\theta}=\mathrm{2}{R}\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)\frac{{d}\omega}{{d}\theta} \\ $$$$+\mathrm{2}{R}\omega\left(−\mathrm{2sin}\:\theta−\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:\mathrm{tan}\:\varphi−\frac{\mathrm{2cos}^{\mathrm{2}} \:\theta}{\mathrm{cos}^{\mathrm{3}} \:\varphi}\right) \\ $$$${a}_{{B}} =\frac{{dv}_{{B}} }{{dt}}=\omega\frac{{dv}_{{B}} }{{d}\theta} \\ $$$$=\mathrm{2}{R}\omega\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)\frac{{d}\omega}{{d}\theta} \\ $$$$+\mathrm{2}{R}\omega^{\mathrm{2}} \left(−\mathrm{2sin}\:\theta−\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\:\mathrm{tan}\:\varphi−\frac{\mathrm{2cos}^{\mathrm{2}} \:\theta}{\mathrm{cos}^{\mathrm{3}} \:\varphi}\right) \\ $$$${N}\mathrm{cos}\:\varphi={ma}_{{B}} \\ $$$${N}=\mathrm{0}\:\Rightarrow{a}_{{B}} =\mathrm{0} \\ $$$$\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)\frac{{d}\omega}{{d}\theta} \\ $$$$−\omega\left(\mathrm{2sin}\:\theta+\mathrm{cos}\:\theta−\mathrm{sin}\:\theta\:\mathrm{tan}\:\varphi+\frac{\mathrm{2cos}^{\mathrm{2}} \:\theta}{\mathrm{cos}^{\mathrm{3}} \:\varphi}\right)=\mathrm{0}\:\:\:..\left({I}\right) \\ $$$$ \\ $$$${x}_{{G}} −{x}_{{A}} =\left(−\mathrm{2sin}\:\theta+\mathrm{cos}\:\theta\right){R} \\ $$$${y}_{{G}} =\mathrm{2}{R}\:\mathrm{cos}\:\theta+{R}\:\mathrm{sin}\:\theta \\ $$$${y}_{{E}} =\mathrm{4}{R}\:\mathrm{cos}\:\theta \\ $$$${I}_{{E}} ={I}+{m}\left({x}_{{E}} −{x}_{{G}} \right)^{\mathrm{2}} +{m}\left({y}_{{E}} −{y}_{{G}} \right)^{\mathrm{2}} \\ $$$$=\frac{{m}\left(\mathrm{4}{R}^{\mathrm{2}} +\mathrm{16}{R}^{\mathrm{2}} \right)}{\mathrm{12}}+{m}\left(−\mathrm{2}\:\mathrm{sin}\:\theta+\mathrm{cos}\:\theta\right)^{\mathrm{2}} {R}^{\mathrm{2}} +{m}\left(−\mathrm{2}\:\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\right)^{\mathrm{2}} {R}^{\mathrm{2}} \\ $$$$=\frac{\left(\mathrm{20}−\mathrm{12}\:\mathrm{sin}\:\mathrm{2}\theta\right){mR}^{\mathrm{2}} }{\mathrm{3}} \\ $$$$\frac{{I}_{{E}} \omega^{\mathrm{2}} }{\mathrm{2}}+\frac{{mv}_{{B}} ^{\mathrm{2}} }{\mathrm{2}}={mg}\left(\sqrt{\mathrm{5}}{R}−{y}_{{G}} \right) \\ $$$$\frac{{I}_{{E}} \omega^{\mathrm{2}} }{\mathrm{2}}+\frac{{mv}_{{B}} ^{\mathrm{2}} }{\mathrm{2}}={mgR}\left(\sqrt{\mathrm{5}}−\mathrm{2}\:\mathrm{cos}\:\theta−\mathrm{sin}\:\theta\right) \\ $$$$\frac{\left(\mathrm{20}−\mathrm{12}\:\mathrm{sin}\:\mathrm{2}\theta\right){m}\omega^{\mathrm{2}} {R}^{\mathrm{2}} }{\mathrm{6}}+\frac{\mathrm{4}{mR}^{\mathrm{2}} \left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)^{\mathrm{2}} \omega^{\mathrm{2}} }{\mathrm{2}}={mgR}\left(\sqrt{\mathrm{5}}−\mathrm{2}\:\mathrm{cos}\:\theta−\mathrm{sin}\:\theta\right) \\ $$$$\left[\mathrm{10}−\mathrm{6}\:\mathrm{sin}\:\mathrm{2}\theta+\mathrm{6}\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)^{\mathrm{2}} \right]\omega^{\mathrm{2}} =\frac{\mathrm{3}{g}}{{R}}\left(\sqrt{\mathrm{5}}−\mathrm{2}\:\mathrm{cos}\:\theta−\mathrm{sin}\:\theta\right) \\ $$$$\omega^{\mathrm{2}} =\frac{\mathrm{3}{g}}{{R}}×\frac{\sqrt{\mathrm{5}}−\mathrm{2}\:\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}{\mathrm{10}−\mathrm{6}\:\mathrm{sin}\:\mathrm{2}\theta+\mathrm{6}\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)^{\mathrm{2}} } \\ $$$$\omega=\sqrt{\frac{\mathrm{3}{g}}{{R}}}×{f}\left(\theta\right) \\ $$$${with}\:{f}\left(\theta\right)=\sqrt{\frac{\sqrt{\mathrm{5}}−\mathrm{2}\:\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}{\mathrm{10}−\mathrm{6}\:\mathrm{sin}\:\mathrm{2}\theta+\mathrm{6}\left(\mathrm{2cos}\:\theta−\mathrm{sin}\:\theta−\mathrm{cos}\:\theta\:\mathrm{tan}\:\varphi\right)^{\mathrm{2}} }} \\ $$$$ \\ $$$${we}\:{get}\:{from}\:\left({I}\right) \\ $$$$\theta\approx\mathrm{0}.\mathrm{7296}\approx\mathrm{41}.\mathrm{8}°\:{at}\:{which}\:{N}=\mathrm{0} \\ $$$$ \\ $$$${all}\:{above}\:{is}\:{valid}\:{for}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\leqslant\theta\leqslant\theta_{{max}} \\ $$$$\varphi+\theta_{{max}} =\mathrm{90}° \\ $$$$\mathrm{sin}\:\left(\mathrm{90}°−\theta_{{max}} \right)=\mathrm{2}\:\mathrm{sin}\:\theta_{{max}} −\mathrm{1} \\ $$$$\mathrm{2}\:\mathrm{sin}\:\theta_{{max}} −\mathrm{cos}\:\theta_{{max}} =\mathrm{1} \\ $$$$\mathrm{sin}\:\left(\theta_{{max}} −\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}} \\ $$$$\theta_{{max}} =\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}+\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\approx\mathrm{53}.\mathrm{1}° \\ $$$${N}=\mathrm{0}\:{occours}\:{at}\:\theta\approx\mathrm{41}.\mathrm{8}°,\:{before} \\ $$$$\theta_{{max}} \:{is}\:{reached}. \\ $$
Commented by mr W last updated on 14/Nov/21
Commented by mr W last updated on 18/Nov/21
$${i}\:{have}\:{taken}\:{following}\:{start}\:{position}: \\ $$$$\theta_{\mathrm{0}} =\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by mr W last updated on 14/Nov/21