Question Number 80648 by mr W last updated on 05/Feb/20
Commented by mr W last updated on 05/Feb/20
$${three}\:{thin}−{wall}\:{tubes}\:{with}\:{identical} \\ $$$${size}\:{and}\:{mass}\:{are}\:{released}\:{from}\:{the} \\ $$$${state}\:{as}\:{shown}. \\ $$$${find}\:{the}\:{time}\:{which}\:{the}\:{upper}\:{tube} \\ $$$${takes}\:{to}\:{hit}\:{the}\:{ground}. \\ $$$${there}\:{is}\:{no}\:{friction}\:{between}\:{the}\:{tubes}, \\ $$$${but}\:{the}\:{ground}\:{is}\:{rough}\:{such}\:{that}\:{the} \\ $$$${tubes}\:{can}'{t}\:{slip}\:{over}\:{it}. \\ $$
Commented by ajfour last updated on 05/Feb/20
Commented by ajfour last updated on 05/Feb/20
$${V}=−\frac{{dy}}{{dt}}\:,\:\:{A}=\frac{{dV}}{{dt}} \\ $$$${y}=\mathrm{2}{r}\mathrm{sin}\:\theta\:\:\Rightarrow\:−{V}=\mathrm{2}{r}\mathrm{cos}\:\theta\left(\frac{{d}\theta}{{dt}}\right) \\ $$$${friction}\:{on}\:{right}\:{tube}\:{is}\:\overset{\leftarrow} {{f}} \\ $$$$\:{fr}=\left({mr}^{\mathrm{2}} \right)\frac{{a}}{{r}}\:\:\:\Rightarrow\:\:{f}={ma} \\ $$$${N}\mathrm{cos}\:\theta−{f}={m}\frac{{dv}}{{dt}}\:=\:{ma} \\ $$$$\Rightarrow\:\:{N}=\frac{\mathrm{2}{ma}}{\mathrm{cos}\:\theta} \\ $$$${mg}−\mathrm{2}{N}\mathrm{sin}\:\theta=\frac{{mVdV}}{{dy}}\:=\:{mA} \\ $$$$\Rightarrow\:\:{mg}−\frac{\mathrm{4}{ma}\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta}={mA} \\ $$$$\Rightarrow\:\:{A}={g}−\mathrm{4}{a}\mathrm{tan}\:\theta \\ $$$${y}=\mathrm{2}{r}\mathrm{sin}\:\theta\:\:,\:\:{x}=\mathrm{2}{r}\mathrm{cos}\:\theta \\ $$$${V}\mathrm{sin}\:\theta={v}\mathrm{cos}\:\theta\:\:\:\:\:\:\:\:\:…….\left({I}\right) \\ $$$${V}=\frac{{v}\mathrm{cos}\:\theta}{\mathrm{sin}\:\theta}=−\mathrm{2}{r}\mathrm{cos}\:\theta\left(\frac{{d}\theta}{{dt}}\right) \\ $$$$\Rightarrow\:\:\:{v}=−\mathrm{2}{r}\mathrm{sin}\:\theta\left(\frac{{d}\theta}{{dt}}\right)\:\:\:\:…..\left({II}\right) \\ $$$${differentiating}\:\left({I}\right) \\ $$$$\Rightarrow\:\:{V}={v}\mathrm{cot}\:\theta \\ $$$$\Rightarrow\:\:{A}\mathrm{sin}\:\theta+{V}\mathrm{cos}\:\theta\frac{{d}\theta}{{dt}}\:=\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\mathrm{cos}\:\theta−{v}\mathrm{sin}\:\theta\frac{{d}\theta}{{dt}} \\ $$$$\Rightarrow\:{g}\mathrm{sin}\:\theta−\mathrm{4}{a}\mathrm{sin}\:\theta\mathrm{tan}\:\theta \\ $$$$\:\:\:\:+{v}\mathrm{cos}\:\theta\mathrm{cot}\:\theta\left(\frac{{d}\theta}{{dt}}\right)\:\:=\: \\ $$$$\:\:\:\:\:\:\:\:{a}\mathrm{cos}\:\theta−{v}\mathrm{sin}\:\theta\left(\frac{{d}\theta}{{dt}}\right) \\ $$$$\Rightarrow\:{g}\mathrm{sin}\:\theta−\mathrm{4}\left(\frac{\mathrm{sin}\:^{\mathrm{2}} \theta}{\mathrm{cos}\:\theta}\right)\frac{{dv}}{{dt}} \\ $$$$+\frac{{v}\mathrm{cos}\:^{\mathrm{2}} \theta}{\mathrm{sin}\:\theta}\left(\frac{{d}\theta}{{dt}}\right)=\frac{{dv}}{{dt}}\mathrm{cos}\:\theta−{v}\mathrm{sin}\:\theta\frac{{d}\theta}{{dt}} \\ $$$${using}\:\left({II}\right) \\ $$$${mgr}\sqrt{\mathrm{3}}−\mathrm{2}{mgr}\mathrm{sin}\:\theta \\ $$$$\:\:\:\:=\:\mathrm{2}{mv}^{\mathrm{2}} +\frac{{mV}^{\:\mathrm{2}} }{\mathrm{2}} \\ $$$${gr}\sqrt{\mathrm{3}}−\mathrm{2}{gr}\mathrm{sin}\:\theta=\mathrm{2}{v}^{\mathrm{2}} +\frac{{v}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \theta}{\mathrm{2sin}\:^{\mathrm{2}} \theta} \\ $$$${v}^{\mathrm{2}} =\frac{\mathrm{2}{gr}\left(\sqrt{\mathrm{3}}−\mathrm{2sin}\:\theta\right)\mathrm{sin}\:^{\mathrm{2}} \theta}{\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} \theta} \\ $$$$\Rightarrow\:\mathrm{2}{v}\frac{{dv}}{{dt}}=\mathrm{2}{gr}\left\{\frac{\left(\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} \theta\right)\left(\mathrm{2}\sqrt{\mathrm{3}}\mathrm{sin}\:\theta\mathrm{cos}\:\theta−\mathrm{6sin}\:^{\mathrm{2}} \theta\mathrm{cos}\:\theta\right)−\mathrm{2sin}\:\theta\mathrm{cos}\:\theta\left(\sqrt{\mathrm{3}}−\mathrm{2sin}\:\theta\right)\mathrm{sin}\:^{\mathrm{2}} \theta}{\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \theta\right)^{\mathrm{2}} }\right\} \\ $$$${a}=\frac{{dv}}{{dt}}=\:\mathrm{0}\:\:\Rightarrow\:{N}=\mathrm{0}\:\: \\ $$$$\Rightarrow\: \\ $$$$\left(\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} \theta\right)\left(\mathrm{2}\sqrt{\mathrm{3}}\mathrm{cos}\:\theta−\mathrm{6sin}\:\theta\mathrm{cos}\:\theta\right)=\mathrm{2cos}\:\theta\left(\sqrt{\mathrm{3}}−\mathrm{2sin}\:\theta\right)\mathrm{sin}\:^{\mathrm{2}} \theta \\ $$$$\theta\:\approx\:\mathrm{33}.\mathrm{92046}° \\ $$$$….. \\ $$
Commented by mr W last updated on 05/Feb/20
$${super}! \\ $$
Commented by mr W last updated on 06/Feb/20
$${final}\:{equation}\:{can}\:{be}\:{simplified}\:{to}: \\ $$$$\mathrm{sin}^{\mathrm{3}} \:\theta+\mathrm{9}\:\mathrm{sin}\:\theta−\mathrm{3}\sqrt{\mathrm{3}}=\mathrm{0} \\ $$$$\mathrm{sin}\:\theta=\sqrt[{\mathrm{3}}]{\frac{\mathrm{3}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{5}}+\mathrm{1}\right)}{\mathrm{2}}}−\sqrt[{\mathrm{3}}]{\frac{\mathrm{3}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{5}}−\mathrm{1}\right)}{\mathrm{2}}} \\ $$$$\Rightarrow\theta=\mathrm{33}.\mathrm{920456}° \\ $$
Commented by ajfour last updated on 06/Feb/20
Commented by ajfour last updated on 06/Feb/20
$${loss}\:{in}\:{potential}\:{energy}\:{is} \\ $$$${mg}\left({r}\sqrt{\mathrm{3}}−\mathrm{2}{r}\mathrm{sin}\:\theta\right) \\ $$$${shouldn}'{t}\:{it}\:{be}\:{so},\:{Sir}\:? \\ $$
Commented by mr W last updated on 06/Feb/20
$${certainly}\:{so}.\:{i}\:{had}\:{a}\:{short}\:{circuit}. \\ $$$${sorry}! \\ $$
Commented by zainal tanjung last updated on 06/Feb/20
$$ \\ $$$$ \\ $$