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 $\mathsf{\text{'none of these'}}$.
