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Question Number 29803 by ajfour last updated on 12/Feb/18

Commented by ajfour last updated on 12/Feb/18

Commented by ajfour last updated on 12/Feb/18

$$\overline{\mathit{\tau }}=\overline{\mathit{r}}\times M\overline{\mathit{g}}$$$$=[R(\cos \mathit{\varphi }\hat{i}+\sin \mathit{\varphi }\hat{j})+\left(R\cot \alpha \right)\hat{k}]\times (-Mg\hat{k})$$$$\overline{\mathit{L}}=I\omega \sin \alpha (\cos \mathit{\varphi }\hat{i}+\sin \mathit{\varphi }\hat{j})$$$$+\left(I\omega \cos \alpha \right)\hat{k}$$$$from\overline{\mathit{\tau }}=\frac{d\overline{L}}{dt},weget$$$$MgR(-\sin \mathit{\varphi }\hat{i}+\cos \mathit{\varphi }\hat{j})=$$$$I\omega \sin \alpha (-\sin \mathit{\varphi }\hat{i}+\cos \mathit{\varphi }\hat{j})\frac{d\mathit{\varphi }}{dt}$$$$\Rightarrow \frac{d\mathit{\varphi }}{dt}=\mathrm{\Omega }=\frac{MgR}{I\omega \sin \alpha }$$$$or\overline{\mathbf{\Omega }}=\frac{MgR}{\left(\frac{{Mr}^2\omega \sin \alpha }{2}\right)}\hat{k}$$$$\overline{\mathbf{\Omega }}=\frac{2gR}{\omega r^2\sin \alpha }\hat{k}$$$$\Rightarrow \overline{\mathit{\tau }}=\overline{\mathbf{\Omega }}\times \mathit{I}\overline{\mathit{\omega }}$$$$MgR=\left(\frac{2gR}{\omega r^2\sin \alpha }\right)\left(\frac{{Mr}^2}{2}\right)\omega \sin \alpha$$$$Thedirectionevenisconsistent.$$

Commented by ajfour last updated on 13/Feb/18

$$Find\mathrm{\Omega }intermsofM,r,R,\omega ,\alpha$$$$given\alpha staysconstant.$$$$Whatis\theta =\alpha \left(t\right),\mathrm{\Omega }\left(t\right)intermsof$$$$\frac{d\omega }{dt}?$$

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