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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 $\small\mathsf{(F=mg)}$, on ideal string wrapped around the inner cylinder (Shown in the figure).

Find the minimum Co-efficient of Friction $\small\mathsf{(\mu_{k})}$ required for Pure Rolling.

My Work

For Pure Rolling, $$\mathsf{ω.r=v \tag 1}$$ $$\mathsf{α.r=a\tag 2}$$

Two torques are acting on the Spool about it's center (Circle's center), one due to the applied force F and other due to the friction generated by the rolling of the spool on the rough horizontal surface. $$\mathsf{∑τ=2r.f_k−F.r=Iα \tag 3}$$ Translation motion's equation, $$\mathsf{\sum F_{ext} = F−f_k=Ma\tag 4}$$ Solving equations $(2), (3)$ and $(4)$ and using $\mathsf{μ_k=\frac{f_k}{N}}$. I am getting $\mathsf{μ=\frac{3}{5}}$. While the options are $\mathsf{\frac{2}{9},\frac{4}{9},\frac{5}{9}}$and $\mathsf{\text{'none of these'}}$.

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closed as too localized by David Z Jan 19 '12 at 4:13

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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Hi Ishaan - this isn't a homework help site, and accordingly it's not the place to get people to just check your work. If you have a conceptual physics question related to your homework problem, you can of course edit this to ask it and I'll be happy to reopen the question. – David Z Jan 19 '12 at 4:17
    
If the pulling force is applied below the CM of the spool, as the diagram suggests, then pure rolling is impossible. The force is to the right so the spool must move right, but if there is pure rolling then the spool would have to wind itself onto the thread, instead of the thread unwinding as it is pulled. – sammy gerbil May 20 at 0:15

Your work looks fine to me. I'd choose "none of these".

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