A cylinder of radius \({R}\) and mass \({M}\) rolls without slipping down a plane inclined at an angle \(\theta\). Coeff. of friction between the cylinder and the plane is \(\mu\). For what maximum inclination \(\theta\), the cylinder rolls without slipping ?

(A) \({{\tan}^{{-{1}}}\mu}\)

(B) \({{\tan}^{{-{1}}}{\left({3}\mu\right)}}\)

(C) \({{\tan}^{{-{1}}}{2}}\mu\)

(D) \({{\tan}^{{-{1}}}.}\frac{{{3}}}{{{2}}}\mu\)

(A) \({{\tan}^{{-{1}}}\mu}\)

(B) \({{\tan}^{{-{1}}}{\left({3}\mu\right)}}\)

(C) \({{\tan}^{{-{1}}}{2}}\mu\)

(D) \({{\tan}^{{-{1}}}.}\frac{{{3}}}{{{2}}}\mu\)

Answer: B

As is know from theory,

\(\mu=\frac{{{\tan{\theta}}}}{{{1}+{m}{R}^{{{2}}}/{I}}}=\frac{{{\tan{\theta}}}}{{{1}+{2}}}=\frac{{{\tan{\theta}}}}{{{3}}}\)

\({{\tan{\theta}}_{{\max}}=}{3}\mu\)

\(\theta_{{\max}}={{\tan}^{{-{1}}}{\left({3}\mu\right)}}\)

As is know from theory,

\(\mu=\frac{{{\tan{\theta}}}}{{{1}+{m}{R}^{{{2}}}/{I}}}=\frac{{{\tan{\theta}}}}{{{1}+{2}}}=\frac{{{\tan{\theta}}}}{{{3}}}\)

\({{\tan{\theta}}_{{\max}}=}{3}\mu\)

\(\theta_{{\max}}={{\tan}^{{-{1}}}{\left({3}\mu\right)}}\)

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