## anonymous 5 years ago $\large\mathsf{\text{Problem Based on Rotational Motion}}$

1. anonymous

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.

2. anonymous

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 $$\large\mathsf{\frac{2}{9}, \frac{4}{9}, \frac{5}{9} \text{and 'none of these'}}$$.

3. anonymous

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.

4. anonymous

Oh yeah, Thank You so much! I knew I did something silly.

5. anonymous

Did you get the correct answer?

6. anonymous

Yeah, I think so $$\Large\mathsf{\mu_{k} =\frac{5}{9}}$$. Thanks!

7. anonymous

Excellent.