${\displaystyle {\vec {\omega }}_{01}={\dot {\phi }}\,{\vec {k}}_{0},\qquad {\vec {v}}_{01}^{\,O}={\vec {0}}.}$

${\displaystyle {\overrightarrow {OG}}=s\,{\vec {\imath }}_{0}+d\cos {\theta }\,{\vec {\jmath }}_{0}+d\,\mathrm {sen} \,\theta \,{\vec {k}}_{0},}$

${\displaystyle {\vec {v}}_{01}^{\,G}={\vec {v}}_{01}^{\,O}+{\vec {\omega }}_{01}\times {\overrightarrow {OG}}=-d{\dot {\phi }}\cos \theta \,{\vec {\imath }}_{0}+s{\dot {\phi }}\,{\vec {\jmath }}_{0}.}$

${\displaystyle {\vec {\omega }}_{20}={\dot {\theta }}\,{\vec {\imath }}_{0},\qquad {\vec {v}}_{01}^{\,A}={\dot {s}}\,{\vec {\imath }}_{0}.}$

${\displaystyle {\overrightarrow {AG}}=d\cos {\theta }\,{\vec {\jmath }}_{0}+d\,\mathrm {sen} \,\theta \,{\vec {k}}_{0},}$

${\displaystyle {\vec {v}}_{20}^{\,G}={\dot {s}}\,{\vec {\imath }}_{0}-d{\dot {\theta }}\,\mathrm {sen} \,\theta \,{\vec {\jmath }}_{0}+d{\dot {\theta }}\cos \theta \,{\vec {k}}_{0}.}$

${\displaystyle {\begin{array}{l}{\vec {\omega }}_{21}={\vec {\omega }}_{20}+{\vec {\omega }}_{01}={\dot {\theta }}\,{\vec {\imath }}_{0}+{\dot {\phi }}\,{\vec {k}}_{0}.\\\\{\vec {v}}_{21}^{\,G}={\vec {v}}_{20}^{\,G}+{\vec {v}}_{01}^{\,G}=({\dot {s}}-d{\dot {\phi }}\cos \theta )\,{\vec {\imath }}_{0}-d{\dot {\theta }}\,\mathrm {sen} \,\theta \,{\vec {\jmath }}_{0}+d{\dot {\theta }}\cos \theta \,{\vec {k}}_{0}.\end{array}}}$

${\displaystyle {\vec {L}}_{G}={\overset {\leftrightarrow }{I}}_{G}\cdot {\vec {\omega }}_{21}.}$

${\displaystyle {\overset {\leftrightarrow }{I}}_{G}\left[{\begin{array}{ccc}I_{0}&0&0\\0&I_{0}&0\\0&0&0\end{array}}\right]_{2}\qquad \qquad I_{0}={\dfrac {1}{12}}m(2d)^{2}={\dfrac {1}{3}}md^{2}.}$

${\displaystyle {\begin{array}{l}{\vec {\imath }}_{0}={\vec {\imath }}_{2},\\{\vec {\jmath }}_{0}=\mathrm {sen} \,\theta \,{\vec {\jmath }}_{2}+\cos \theta \,{\vec {k}}_{2},\\{\vec {k}}_{0}=-\cos \theta \,{\vec {\jmath }}_{2}+\mathrm {sen} \,\theta \,{\vec {k}}_{2}.\end{array}}}$

${\displaystyle {\vec {\omega }}_{21}={\dot {\theta }}\,{\vec {\imath }}_{0}+{\dot {\phi }}\,{\vec {k}}_{0}={\dot {\theta }}\,{\vec {\imath }}_{2}-{\dot {\phi }}\cos \theta \,{\vec {\jmath }}_{0}+{\dot {\phi }}\,\mathrm {sen} \,\theta \,{\vec {k}}_{2}.}$

${\displaystyle {\vec {L}}_{G}=\left[{\begin{array}{ccc}I_{0}&0&0\\0&I_{0}&0\\0&0&0\end{array}}\right]_{2}\left[{\begin{array}{c}{\dot {\theta }}\\-{\dot {\phi }}\cos \theta \\{\dot {\phi }}\,\mathrm {sen} \,\theta \end{array}}\right]_{2}=\left[{\begin{array}{c}I_{0}{\dot {\theta }}\\-I_{0}{\dot {\phi }}\cos \theta \\0\end{array}}\right]_{2}}$

${\displaystyle {\begin{array}{l}{\vec {\omega }}_{21}=[{\dot {\theta }},\,0,\,0]_{0}\\\\{\vec {v}}_{21}^{\,G}=[{\dot {s}},\,-d{\dot {\theta }}\,\mathrm {sen} \,\theta \,d{\dot {\theta }}\cos \theta ]_{0}\\\\{\vec {L}}_{G}=[I_{0}{\dot {\theta }},\,0,\,0]_{0}.\end{array}}}$

${\displaystyle T=T_{tra}+T_{rot}}$

${\displaystyle T_{tra}={\dfrac {1}{2}}m|{\vec {v}}_{21}^{\,G}|^{2}={\dfrac {1}{2}}m({\dot {s}}^{2}+d^{2}{\dot {\theta }}^{2}).}$

${\displaystyle T_{rot}={\dfrac {1}{2}}{\vec {L}}_{G}\cdot {\vec {\omega }}_{21}={\dfrac {1}{6}}md^{2}{\dot {\theta }}^{2}.}$

${\displaystyle T={\dfrac {1}{6}}m(3{\dot {s}}^{2}+4d^{2}{\dot {\theta }}^{2}).}$

${\displaystyle U_{g}=mgd\,\mathrm {sen} \,\theta ,\qquad U_{k}={\dfrac {1}{2}}k(s-d)^{2}.}$

${\displaystyle U=U_{g}+U_{k}=mgd\,\mathrm {sen} \,\theta +{\dfrac {1}{2}}k(s-d)^{2}.}$

${\displaystyle L=T-U={\dfrac {1}{6}}m(3{\dot {s}}^{2}+4d^{2}{\dot {\theta }}^{2})-mgd\,\mathrm {sen} \,\theta -{\dfrac {1}{2}}k(s-d)^{2}.}$

${\displaystyle \left.{\begin{array}{l}{\dfrac {\partial L}{\partial {\dot {s}}}}=m{\dot {s}},\\\\{\dfrac {\mathrm {d} }{\mathrm {d} t}}\left({\dfrac {\partial L}{\partial {\dot {s}}}}\right)=m{\ddot {s}},\\\\{\dfrac {\partial L}{\partial s}}=-k(s-d).\end{array}}\right\}\to {\ddot {s}}+{\dfrac {k}{m}}(s-d)=0}$

${\displaystyle \left.{\begin{array}{l}{\dfrac {\partial L}{\partial {\dot {\theta }}}}={\dfrac {4}{3}}md^{2}{\dot {\theta }},\\\\{\dfrac {\mathrm {d} }{\mathrm {d} t}}\left({\dfrac {\partial L}{\partial {\dot {\theta }}}}\right)={\dfrac {4}{3}}md^{2}{\ddot {\theta }},\\\\{\dfrac {\partial L}{\partial \theta }}=-mgd\cos \theta .\end{array}}\right\}\to {\ddot {\theta }}+{\dfrac {3g}{4d}}\cos \theta =0.}$

${\displaystyle s(0)=d,\qquad {\dot {s}}(0)=0,\qquad \theta (0)=-\pi /2,\qquad {\dot {\theta }}(0)=0.}$

${\displaystyle {\vec {v}}_{21}^{\,B}={\vec {v}}_{21}^{\,G}+{\vec {\omega }}_{21}\times {\overrightarrow {GB}}=[{\dot {s}},-2d{\dot {\theta }}\,\mathrm {sen} \,\theta ,2d{\dot {\theta }}\cos \theta ]_{0}.}$

${\displaystyle p_{s}={\dfrac {\partial L}{\partial {\dot {s}}}}=m{\dot {s}},\qquad p_{\theta }={\dfrac {\partial L}{\partial {\dot {\theta }}}}={\dfrac {4}{3}}md^{2}{\dot {\theta }}.}$

${\displaystyle {\begin{array}{l}\Delta p_{s}={\hat {Q}}_{s}^{NC}=\left.{\vec {\hat {F}}}\cdot {\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {s}}}}\right|_{t=0},\\\\\Delta p_{\theta }={\hat {Q}}_{s}^{NC}=\left.{\vec {\hat {F}}}\cdot {\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {\theta }}}}\right|_{t=0}.\end{array}}}$

${\displaystyle \Delta p_{s}=m{\dot {s}}^{+},\qquad \Delta p_{\theta }={\dfrac {4}{3}}md^{2}{\dot {\theta }}^{+}.}$

${\displaystyle {\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {s}}}}=[1,\,0,\,0]_{0},\qquad {\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {\theta }}}}=[0,\,-2d\,\mathrm {sen} \,\theta ,\,2d\cos \theta ]_{0}.}$

${\displaystyle \left.{\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {s}}}}\right|_{t=0}=[1,\,0,\,0]_{0},\qquad \left.{\dfrac {\partial {\vec {v}}_{21}^{\,B}}{\partial {\dot {\theta }}}}\right|_{t=0}=[0,\,2d,\,0]_{0}.}$

${\displaystyle Q_{s}^{NC}={\hat {F}}_{0},\qquad Q_{\theta }^{NC}=0.}$

${\displaystyle {\dot {s}}(0^{+})={\dfrac {{\hat {F}}_{0}}{m}},\qquad {\dot {\theta }}(0^{+})=0.}$