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Question Number 105594 by ajfour last updated on 30/Jul/20

Commented by ajfour last updated on 30/Jul/20

$$Findmaximum\mathit{x},fory>0.$$$$Thetwocylindershaveradii\mathit{r}.$$

Commented by ajfour last updated on 30/Jul/20

Commented by ajfour last updated on 30/Jul/20

$$s=f=g$$$$letscallmg=w$$$$s\cos \theta +N\sin \theta +mg=R=\frac{g}{\mu }$$$$g+s\sin \theta =N\cos \theta$$$$F=g$$$$R+f=2mg$$$$N\sin \theta +s\cos \theta +f=mg$$$$---------------$$$$\Rightarrow g\cos \theta +\frac{g(1+\sin \theta )\sin \theta +w}{\cos \theta }=\frac{g}{\mu }$$$$\frac{g}{\mu }+g=2w\Rightarrow \frac{w}{g}=\frac{1}{2}(1+\frac{1}{\mu })$$$$\frac{g(1+\sin \theta )\sin \theta }{\cos \theta }+g\cos \theta +g=w$$$$\Rightarrow \frac{1}{2}(1+\frac{1}{\mu })=(\frac{1}{\mu }-\cos \theta )\cos \theta$$$$-\sin \theta (1+\sin \theta )$$$$=1+\cos \theta +\frac{\sin \theta }{\cos \theta }(1+\sin \theta )$$$$\Rightarrow \frac{1}{\mu }(\cos \theta -\frac{1}{2})=\sin \theta +\frac{3}{2}$$$$\Rightarrow \frac{1}{2}(1+\frac{3+2\sin \theta }{2\cos \theta -1})=1+\frac{1+\sin \theta }{\cos \theta }$$$$\Rightarrow 2\cos \theta -1=\cos \theta$$$$\Rightarrow \cos \theta =1$$$$\Rightarrow y=0.$$

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