Question Number 82806 by ajfour last updated on 24/Feb/20
Commented by ajfour last updated on 24/Feb/20
$$\mathrm{If}\:\mathrm{released}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{by}\:\mathrm{dashed} \\ $$$$\mathrm{lines},\:\mathrm{subsequently}\:\mathrm{find}\:\theta\left(\mathrm{x}\right). \\ $$$$\mathrm{Assume}\:\mathrm{length}\:\mathrm{of}\:\mathrm{rod}\:\mathrm{L}. \\ $$
Answered by mr W last updated on 24/Feb/20
Commented by mr W last updated on 24/Feb/20
$${a}={g}\:\mathrm{sin}\:\theta \\ $$$${N}={m}\left({g}\:\mathrm{cos}\:\theta−\alpha{x}\right) \\ $$$${Mg}\:\frac{{L}\:\mathrm{cos}\:\theta}{\mathrm{2}}+{Nx}=\frac{{ML}^{\mathrm{2}} }{\mathrm{3}}\alpha \\ $$$${Mg}\:\frac{{L}\:\mathrm{cos}\:\theta}{\mathrm{2}}+{m}\left({g}\:\mathrm{cos}\:\theta−\alpha{x}\right){x}=\frac{{ML}^{\mathrm{2}} }{\mathrm{3}}\alpha \\ $$$$\left(\:\frac{{ML}}{\mathrm{2}}+{mx}\right){g}\:\mathrm{cos}\:\theta=\left(\frac{{ML}^{\mathrm{2}} }{\mathrm{3}}+{mx}^{\mathrm{2}} \right)\alpha \\ $$$$\alpha=\frac{{d}^{\mathrm{2}} \theta}{{dt}^{\mathrm{2}} }=\frac{{g}\left(\:\frac{{ML}}{\mathrm{2}}+{mx}\right)\mathrm{cos}\:\theta}{\frac{{ML}^{\mathrm{2}} }{\mathrm{3}}+{mx}^{\mathrm{2}} } \\ $$$${a}=\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }={g}\:\mathrm{sin}\:\theta \\ $$$${x}\left(\mathrm{0}\right)={c}\:\left({marked}\:{as}\:{a}\:{in}\:{diagram}\right) \\ $$$${v}\left(\mathrm{0}\right)={x}'\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\theta\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\omega\left(\mathrm{0}\right)=\theta'\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${let}\:\mu=\frac{{m}}{{M}},\:\xi=\frac{{x}}{{L}} \\ $$$$\frac{{d}^{\mathrm{2}} \theta}{{dt}^{\mathrm{2}} }=\frac{{g}\left(\:\frac{\mathrm{1}}{\mathrm{2}}+\mu\xi\right)\mathrm{cos}\:\theta}{{L}\left(\frac{\mathrm{1}}{\mathrm{3}}+\mu\xi^{\mathrm{2}} \right)}\:\:\:…\left({i}\right) \\ $$$$\frac{{d}^{\mathrm{2}} \xi}{{dt}^{\mathrm{2}} }=\frac{{g}}{{L}}\:\mathrm{sin}\:\theta\:\:\:…\left({ii}\right) \\ $$$$ \\ $$$${we}\:{solve}\:\left({i}\right)\:{and}\:\left({ii}\right)\:{to}\:{get}\:\xi\left({t}\right)\:{and} \\ $$$$\theta\left({t}\right)\:\left({theoretically}\right) \\ $$$$…… \\ $$
Commented by ajfour last updated on 24/Feb/20