\[\large\mathsf{\text{Problem Based on Rotational Motion}}\]

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\[\large\mathsf{\text{Problem Based on Rotational Motion}}\]

Physics
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A spool of mass \(\mathsf{m}\) and inner radius \(\mathsf{r}\) and outer radius \(\mathsf{2r}\), having moment of inertia \(\Large\mathsf{\frac{mr^2}{2}}\) is made to roll without sliding on a rough horizontal surface by the help of an applied force \(\mathsf {(F = mg)}\), on ideal string wrapped around the inner cylinder (Shown in the figure). |dw:1326937711360:dw| Find the minimum Co-efficient of Friction required for Pure Rolling.
Here's what I did. |dw:1326938034465:dw| For Pure Rolling, \(\mathsf{\omega . r = v \tag{1}}\) \(\mathsf{\alpha . r = a }\tag{2}\) Two torques are acting on the Spool about it's center (Circle's center), one due to the applied force \(\mathsf F\) and other due to the friction generated by the rolling of the spool on the rough horizontal surface. \[\mathsf{\sum \tau = 2r. f_{k} - F.r = I \alpha }\tag 3\] Translation motion's equation, \[\mathsf{ F - f_{k} = Ma} \tag 4\] Solving equations (2), (3) and (4) and using \(\Large\mathsf{ \mu_{k} = \frac{f_{k}}{N}}\). I am getting \(\Large\mu = \frac{3}{5}\). While the options are \(\large\mathsf{\frac{2}{9}, \frac{4}{9}, \frac{5}{9} \text{and 'none of these'}}\).
Double check a few things. First, note that\[\alpha (2r) = a_{CM}\]Also, note that\[\sum \vec {\bf F}_x = 0\]Otherwise, \[\alpha (2r) = a_{CM}\]won't be satisfied.

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Oh yeah, Thank You so much! I knew I did something silly.
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Yeah, I think so \(\Large\mathsf{\mu_{k} =\frac{5}{9}}\). Thanks!
Excellent.

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