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Question Number 184602 by ajfour last updated on 09/Jan/23

Commented by ajfour last updated on 09/Jan/23

$$Ifthesolidsphererestingon$$$$therimofathinhollowfixed$$$$cylinderissetinmotionatan$$$$angularspeed{\omega }_{0}asshown,then$$$$findthetimeittakestocometo$$$$astop.T=f({\omega }_0,a,R,\mu ,g)$$

Answered by mr W last updated on 09/Jan/23

Commented by mr W last updated on 09/Jan/23

$$r=\sqrt{R^2-{(R-a)}^2}=\sqrt{(2R-a)a}$$$$\cos \theta =\frac{R-a}{R}$$$$2\pi rN\cos \theta =Mg$$$$\Rightarrow 2\pi rN=\frac{MgR}{R-a}$$$$f=\mu N$$$$\tau =2\pi rfr=\mu 2\pi rNr=\frac{\mu MgR\sqrt{(2R-a)a}}{R-a}$$$$\alpha I=\tau$$$$\alpha =\frac{\tau }{I}=\frac{\mu MgR\sqrt{(2R-a)a}}{(R-a)\frac{2{MR}^2}{5}}=\frac{5\mu g\sqrt{(2R-a)a}}{2R(R-a)}$$$$T=\frac{{\omega }_0}{\alpha }=\frac{2{\omega }_{0}R(R-a)}{5\mu g\sqrt{(2R-a)a}}✓$$

Commented by ajfour last updated on 09/Jan/23

$$Indeed,beautifulsolutionSir!$$

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