Problem 7.164 Challenge

จาก \(Σ\vec{L}_{f}= \Sigma \vec{L}_{i};\) ขณะชน \(Στ = 0\)

\(|\Sigma \vec{L}_{f}|= |\Sigma \vec{L}_{i}|\)

\(I_{f}\omega = rp_{i}\)

\((m\ell ^{2}+\frac{1}{3}M\ell ^{2})\omega = (\ell )(mu)\)

\(\omega =\frac{3mu}{(3m + M)\ell}\)

จาก \(\Sigma W_{NC}= \Delta E\)

\(0 = \Delta K + \Delta U\)

\(0=-\frac{1}{2}I\omega ^{2}+Mg\frac{\ell}{2}(1-\cos \theta )+mg\ell (1-\cos \theta )\)

\(\cos \theta = 1 -\frac{I}{(2m + M)g\ell}\omega ^{2}\)

\(\cos \theta = 1 -\frac{(m\ell ^{2} +\frac{1}{3}M\ell ^{2})}{(2m + M)g\ell}(\frac{3mu}{(3m + M)\ell})^{2}\)

\(\theta = \cos ^{-1}(1 -\frac{3m^{2}u^{2}}{(3m + M)(2m + M)g\ell}) \;\blacksquare\)