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Question Number 108588 by ajfour last updated on 17/Aug/20

Commented by ajfour last updated on 21/Aug/20

$$Thereisnofrictionbetweenblock$$$$andsphere,butenoughfriction$$$$betweengroundandsphere,such$$$$thatsphererollsasblockslides$$$$backwards.Findmaximum$$$$velocityacquiredbysphere\left(hollow\right).$$

Answered by ajfour last updated on 21/Aug/20

Commented by ajfour last updated on 21/Aug/20

$$\left(N\sin \theta \right)R(1+\cos \theta )=I\alpha ...\left(i\right)$$$$A=\alpha R$$$$mg\cos \theta -N-m\alpha R\sin \theta =\frac{{mv}^2}{R}..\left(ii\right)$$$$mg\sin \theta +m\alpha R\cos \theta =\frac{mvdv}{Rd\theta }...\left(iii\right)$$$$\frac{v}{R}=\frac{d\theta }{dt}...\left(iv\right)$$$$\Rightarrow$$$$mg\sin \theta -\frac{I\alpha }{R\sin \theta (1+\cos \theta )}$$$$-m\alpha R\sin \theta =mR{\left(\frac{d\theta }{dt}\right)}^2$$$$from\left(iii\right)$$$$R\alpha =\frac{1}{\cos \theta }(\frac{vdv}{Rd\theta }-g\sin \theta )$$$$andlet\frac{I}{{mR}^2}=k,then$$$$g\sin \theta -(\frac{k}{\sin \theta (1+\cos \theta )}+\sin \theta )\alpha R$$$$=R{\left(\frac{d\theta }{dt}\right)}^2\Rightarrow$$$$\frac{d^2\theta }{{dt}^2}-\left(\frac{g}{R}\right)\sin \theta =\frac{[\left(\frac{g}{R}\right)\sin \theta -{\left(\frac{d\theta }{dt}\right)}^2]\cos \theta }{\sin \theta +\frac{k}{\sin \theta (1+\cos \theta )}}$$$$......................................................$$

Answered by mr W last updated on 22/Aug/20

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