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Question Number 53468 by ajfour last updated on 22/Jan/19

Commented by ajfour last updated on 22/Jan/19

$$Find\theta intermsof\mathit{h},\mathit{r},and\mathit{R}.$$$$\left(ringrestsinsidecylinder\right).$$

Commented by mr W last updated on 23/Jan/19

$$pleasechecksir:$$$$suchthattheringcanbeplacedinto$$$$thecylinder\Rightarrow R⩽r$$$$butifR⩽r\Rightarrow theringcannotreston$$$$thecylinder.$$$$\Rightarrow nosolution?$$

Commented by ajfour last updated on 23/Jan/19

$$onlypossibilityish\ll randR>r.$$$$yessir,whatyousayistrue.$$

Answered by mr W last updated on 23/Jan/19

$$AB=2b=2r\sin \frac{\theta }{2}$$$$\Rightarrow b=r\sin \frac{\theta }{2}$$$${(r+\sqrt{r^2-b^2})}^2+h^2={(R+\sqrt{R^2-b^2})}^2$$$$\Rightarrow R\sqrt{R^2-b^2}-r\sqrt{r^2-b^2}=\frac{h^2}{2}-(R^2-r^2)$$$$\Rightarrow \frac{R}{r}\sqrt{\frac{R^2}{r^2}-{\sin }^2\frac{\theta }{2}}-\cos \frac{\theta }{2}=\frac{h^2}{2r^2}-\frac{R^2}{r^2}+1$$$$with\mu =\frac{R}{r},\eta =\frac{h}{\sqrt{2}r}$$$$\Rightarrow {\mu }^2+\mu \sqrt{{\mu }^2-1+{\cos }^2\frac{\theta }{2}}-\cos \frac{\theta }{2}=1+{\eta }^2$$$$\Rightarrow \cos \frac{\theta }{2}=....$$

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